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SYMMETRIES of B s and K s A Pedagogical Consideration: Simplification to a homework problem This Month‘s Special RUF‘s Theorem Unitarity of D H.-J. Gerber, ETHZ MITP Workshop on T violation and CPT tests in neutral-meson systems April 2013, Uni Mainz.

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= k = 1, 2, 3. ψ(t) → Homomorphism SL(2,C) onto sLT CPLEAR CP violation T violation ψ( t ) = e -iΛt ψ(0) CONTRADICTS Time reversal symmetry of H weak

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CP is violated, whenever... CPT (δ) and/or T (Re(ε)) is violated x3x3 x2x2 Im(ε-δ) Re(ε-δ) KLKL x1x1 Cannot have CP conserved, T and CPT violated. Reward: Conservation Law for CP.

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The Parameters of Measurement Time evolution governed by „Grand Schödinger Equation“ iħ(∂/∂t) |ψ ALL > = H |ψ ALL > with H = H str+elm + H weak Reduce to 2 dimensions: |ψ ALL > |ψ>. Amplitude for evolution and decay of B: A Bf = Measure T, Λ Reward for CP: (for not having its own parameter) If CP -1 T CP = T and CP + = CP -1 (unitary) then = ( = )

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Measure the matrix T for B s Assumptions Λ is Time reversal- and CPT symmetric, ΔΓ = 0, T i≠j = 0 „Δb = ΔQ rule“, No FSI. Then U = e -iΛt = U 0 Matrix representation of | A Bf | 2 = | | 2 | A Bf | 2 = f i * f j T *ii T jj U *ik U jm b k * b m. Basis K 0 K 0, B 0 B 0. cos(Δm t/2) -i sin(Δm t/2) -i sin(Δm t/2) cos(Δm t/2), |U 0 | 2 = e -Γt. _ _

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Strategies significance and choice Greatest sensitivity of | A Bf | 2 to T. | A Bf | 2 /e -Γt = (1/4) (|T 11 | 2 + |T 22 | 2 )[(f 1 2 +f 2 2 ) (b 1 2 +b 2 2 ) + (f 1 2 -f 2 2 ) (b 1 2 -b 2 2 ) cos(Δm t)]+ (1/4) (|T 11 | 2 - |T 22 | 2 )[(f f 2 2 ) (b 1 2 +b 2 2 ) + (f 1 2 +f 2 2 ) (b 1 2 -b 2 2 ) cos(Δm t)]+ 2 Re(T 11* T 22 ) [ f 1 f 2 b 1 b 2 ]+ Im(T 11* T 22 ) [ f 1 f 2 (b 1 2 -b 2 2 ) sin(Δm t)].(1) Examples: B 0 ~ K 0 ~ (Phase convention) _ _ J/ψK S ~ J/ψK L ~ Shorthands: (|T 11 | 2 - |T 22 | 2 ) ≡ T CPT Im(T 11 * T 22 ) ≡ T T

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CP violation T T ≠ 0 or/and T CPT ≠ 0. (- T CPT cos(mΔ t) – 2 T T sin(mΔ t) ) / ( |T 11 | 2 + |T 22 | 2 ). From (1). sin(mΔ t) T T found Aubert et. al, (BABAR Collaboration) Observation of CP violation in the B 0 meson system Phys. Rev. Lett. 87, (2001). Data show Sine ! No sign of cosine ? Is the question Fourier analysis by eye

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CPT violation _ Calculate Fwd = Bwd = Find ( |Fwd| 2 - |Bwd| 2 ) /e -Γt = - T CPT cos(Δm t). Compare- T CPT cos(Δm t)CPT - T CPT cos(Δm t) – 2 T T sin(Δm t)CP Experiments T violation ! DIRECT ?

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T violation and Motion Reversal Motion Reversal: Compare B 0 B + vs B + B 0. Identify the B + (~ K L ) by observing the decay to K S („first decay“) of its orthogonal entangled partner (B - ). Let |h> = state of first decay’s products. The surviving state is |surv> = iσ 2 K T -1 |h>. σ 2 : Pauli matrix, K : Complex conjugation. The backward amplitude Bwd is then Bwd =, to compare with Fwd =. Need a matrix representation of | Bwd | 2 = | | 2 “Disentanglement Operator” D ≡ iσ 2 K T -1.

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MR violation, is it T violation ? | Bwd | 2 |T 11 T 22 | 2 /e -Γt = | | 2 |T 11 T 22 | 2 /e -Γt = (1/4) (|T 11 | 2 + |T 22 | 2 )[(h 1 2 +h 2 2 ) (b 1 2 +b 2 2 ) + (h h 2 2 ) (b 1 2 -b 2 2 ) cos(Δm t)] + (1/4) T CPT [(h h 2 2 ) (b 1 2 +b 2 2 ) + (h 1 2 +h 2 2 ) (b 1 2 -b 2 2 ) cos(Δm t)] + 2 Re(T *11 T 22 ) [ h 1 h 2 b 1 b 2 ] + T T [ h 1 h 2 (b 1 2 -b 2 2 ) sin(Δm t)].(2) Assumption |T 11 T 22 | 2 = 1 (preliminary). Apply (1) and (2) to B 0 B + vs B + B 0. |f > = |K L >, |h> = |K S >. MRV (B 0 B + ) = ( |Fwd| 2 - |Bwd| 2 ) /e -Γt = T CPT cos(Δm t) – 2 T T sin(Δm t).(3) Compare CP violation - T CPT cos(mΔ t) – 2 T T sin(mΔ t). NO ? ( sometimes YES )

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RUF‘s Theorem Background Bernabeu, Martinez-Vidal, and Villanueva-Perez, JHEP 08 (2012) 064 F. Martinez-Vidal (CERN EP Seminar 2012) T. Ruf This implies T CPT = 0. MRV, eq.(3), is insensitive to CPTV. „B + and B +, and B_ and B_ have to be the same states.“ „If the surviving state B_ needs to be the same as the state B_, then |T 11 | = |T 22 |.” ~~ ~

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Another Proof (unitarity) Ruf‘s Theorem follows also as a special case from the physical requirement, that the Disentanglement Operator D be unitary. This entails unitarity of T. With the „Δb = ΔQ rule“, T 12 = T 21 = 0, the result follows. Let D + = D -1. D = iσ 2 T -1*. Note: iσ 2 = real, orthogonal, anti-symmetric, non-singular. Thus, T is also unitary: T + = T -1. E. g. |T 11 | 2 + |T 21 | 2 = |T 22 | 2 + |T 12 | 2 =1 T 11 T 12 * + T 21 T 22 * = 0. Corollary (RUF) Apply “Δb = Q rule“ T 12 = T 21 = 0 and find |T 11 | = |T 22 | ( = 1 ).

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Apply Unitarity of D CPT violation: MRV (B 0 B_ vs B_ B 0 ) = ( |Fwd| 2 - |Bwd| 2 ) /e -Γt = -T CPT cos(Δm t) ≡ 0. Motion Reversal together with D isentanglement excludes testing of CPT violation, but fine for T T. _ CPT violation: MRV (B 0 B_ vs B_ B 0 ) = ( |Fwd| 2 - |Bwd| 2 ) /e -Γt = -

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Link to Basic Physics, Summary The symmetry properties of H weak in the „Grand Schrödinger Equation“ (Significance for Basic Physics ) require for the parameters in the 2 dimensional representation: This means, in the model of the „homework problem“ Experiment Compatible with symmetry in H weak Incompatible with symmetry in H weak CPTT, CP MRV „ T “ CPT chosenT, (CP) MRV „CPT“ CPT chosennone

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GREAT THANKS TO MARIA FIDECARO CERN THOMAS RUF CERN and YOU !

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FAQ s 1Two Amplitudes in One Channel 2Selected References

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FAQ 1

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FAQ 2 Selected References Kaons on the globe (and 121 references) The fundamental symmetries in the neutral kaon system – a pedagogical choice Maria Fidecaro and Hans-Jürg Gerber, Rep. Prog. Phys.69 (2006) Significance of the parameters for H weak (and much on fundamentals) On the phenomenological description of CP violation for K-mesons and its consequences C. P. Enz and R. R. Lewis, Helv. Phys. Acta 38 (1965) Reprinted in L. Wolfenstein (ed), CP Violation, North-Holland (1989).

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