Constraining CP violation in neutral meson mixing with theory input

Transcription

1 Constraining CP violation in neutral meson mixing with theory input Sascha Turczyk Work in collaboration with M. Freytsis and Z. Ligeti [ hep-ph] Lawrence Berkeley National Laboratory Vienna Theory Seminar Thursday, June 28th, 2012 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 1 / 30

2 Outline 1 Introduction Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters 2 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data 3 Discussuion Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 2 / 30

3 Motivation Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters The Standard Model has passed all precision tests 1 CERN: Z discovery, test of the gauge structure 2 Flavour factories: Test of the flavour sector 3 Tevatron: Discoveries, top, B s B s oscillation,... 4 LHC: Up to now no significant new discoveries Only a few tensions 2 3σ Most hints for New Physics in flavour physics sector Promising Channels: Flavour changing neutral currents (FCNC) Forbidden at tree level NP can enter at the same order F = 1 processes: Rare decays F = 2 processes: Meson oscillation / mixing Focus here on M M oscillation (especially B d/s B d/s ) Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 3 / 30

4 Motivation Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters The Standard Model has passed all precision tests 1 CERN: Z discovery, test of the gauge structure 2 Flavour factories: Test of the flavour sector 3 Tevatron: Discoveries, top, B s B s oscillation,... 4 LHC: Up to now no significant new discoveries Only a few tensions 2 3σ Most hints for New Physics in flavour physics sector Promising Channels: Flavour changing neutral currents (FCNC) Forbidden at tree level NP can enter at the same order F = 1 processes: Rare decays F = 2 processes: Meson oscillation / mixing Focus here on M M oscillation (especially B d/s B d/s ) Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 3 / 30

5 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Mixing and CP Violation: Origin and Consequences CKM Matrix Diagonalize up- and down-type quark mass matrices simultaneously Missmatch in charged current described by CKM matrix V ckm = V (u) W (d) 3 generations V ckm has 3 angles and 1 complex phase Consequences CP violation if all masses are non-degenerate Transitions between different generations Flavor changing neutral currents at the loop-level Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 4 / 30

6 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Mixing and CP Violation: Origin and Consequences CKM Matrix Diagonalize up- and down-type quark mass matrices simultaneously Missmatch in charged current described by CKM matrix V ckm = V (u) W (d) 3 generations V ckm has 3 angles and 1 complex phase Consequences CP violation if all masses are non-degenerate Transitions between different generations Flavor changing neutral currents at the loop-level Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 4 / 30

7 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Mixing and CP Violation: Origin and Consequences CKM Matrix Diagonalize up- and down-type quark mass matrices simultaneously Missmatch in charged current described by CKM matrix V ckm = V (u) W (d) 3 generations V ckm has 3 angles and 1 complex phase Consequences CP violation if all masses are non-degenerate Transitions between different generations Flavor changing neutral currents at the loop-level Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 4 / 30

8 The CKM Matrix Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters 1.5 excluded at CL > 0.95 excluded area has CL > γ & m s m d sin 2β 0.5 ε K α m d η α V ub SL V ub τ ν γ β α -1.0 CKM f i t t e r Winter 12 γ ε K sol. w/ cos 2β < 0 (excl. at CL > 0.95) [CKMfitter: Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 5 / 30 ρ

9 CP Violating in Mixing Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters General Comments [Bigi,Sanda: CP violation] Occurs in P 0 P 0 oscillations Flavor specific final states P 0 f P 0 Neccessary condition: 1 q p Rates for B and B differ Example of Process Semi-leptonic asymmetry: P 0 X l + ν l and P 0 X l ν l A sl Γ (P 0 X l ) Γ ( P 0 X l + ) Γ (P 0 X l ) + Γ ( P 0 X l + ) = 1 q/p q/p 4 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 6 / 30

10 Description of Neutral Meson Mixing Two state system with interplay of oscillation and decay Mass matrix M and decay width matrix Γ are hermitian ( ) [( ) ( )] ( i P 0 M 11 M 12 = t P 0 M12 i Γ 11 Γ 12 P 0 M 11 2 Γ12 Γ 11 P 0 Diagonalization Solution for mass eigenstates P H,L = p P0 q P 0 p 2 + q 2, Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters s s ) b b q 2 p 2 = 2M 12 iγ 12 2M 12 iγ 12 Mass eigenstates do not need to coincide with CP eigenstates δ P H P L = p 2 q 2 p 2 + q = 1 q/p q/p = 1 1 A 2 sl A sl A sl b s b s Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 7 / 30

11 Description of Neutral Meson Mixing Two state system with interplay of oscillation and decay Mass matrix M and decay width matrix Γ are hermitian ( ) [( ) ( )] ( i P 0 M 11 M 12 = t P 0 M12 i Γ 11 Γ 12 P 0 M 11 2 Γ12 Γ 11 P 0 Diagonalization Solution for mass eigenstates P H,L = p P0 q P 0 p 2 + q 2, Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters s s ) b b q 2 p 2 = 2M 12 iγ 12 2M 12 iγ 12 Mass eigenstates do not need to coincide with CP eigenstates δ P H P L = p 2 q 2 p 2 + q = 1 q/p q/p = 1 1 A 2 sl A sl A sl b s b s Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 7 / 30

12 Test of the Standard Model Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Need to predict three parameters to compare with SM s b s b b s b M = 2Re (M 12 i/2γ 12)(M12 i/2γ 12 ) 2 M12 Γ = 4Im (M 12 i/2γ 12)(M12 i/2γ 12 ) 2 Γ12 cos[arg( Γ12/M12)] δ = (1 q/p 2 )/(1 + q/p 2 ) 1/2 Im Γ 12/M 12 Mixing Parameter Input [ , ] M 12 : Dominated by dispersive part of B = 2 operator Γ 12 : Dominated by absorpative part of B = 1 op. double insertion Main theoretical uncertainties 1 Operator product expansion in physical region 2 Expansion in small energy release m b 2m c < 2 GeV s Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 8 / 30

13 Test of the Standard Model Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Need to predict three parameters to compare with SM s b s b b s b M = 2Re (M 12 i/2γ 12)(M12 i/2γ 12 ) 2 M12 Γ = 4Im (M 12 i/2γ 12)(M12 i/2γ 12 ) 2 Γ12 cos[arg( Γ12/M12)] δ = (1 q/p 2 )/(1 + q/p 2 ) 1/2 Im Γ 12/M 12 Mixing Parameter Input [ , ] M 12 : Dominated by dispersive part of B = 2 operator Γ 12 : Dominated by absorpative part of B = 1 op. double insertion Main theoretical uncertainties 1 Operator product expansion in physical region 2 Expansion in small energy release m b 2m c < 2 GeV s Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 8 / 30

14 Introduction Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Effective Theory at the scale of the B X 4GF Ci (µ)oi (µ) Heff = λckm 2 i Current-current operators Electroweak/QCD Penguins Magnetic Penguins Semi-leptonic operators F = 2 operators Allows for Systematic Calculation: Heavy Quark Expansion (HQE) Perturbative αs corrections Non-perturbative 1/mb,c corrections Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 9 / 30

15 Introduction Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Effective Theory at the scale of the B X 4GF Ci (µ)oi (µ) Heff = λckm 2 i Current-current operators Electroweak/QCD Penguins Magnetic Penguins Semi-leptonic operators F = 2 operators Allows for Systematic Calculation: Heavy Quark Expansion (HQE) Perturbative αs corrections Non-perturbative 1/mb,c corrections Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 9 / 30

16 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters The Hamilton Matrix: Computing Mixing Parameters Matrix to be understood in M 1 M M M 2 space Weak interaction sets scale: Wigner-Weisskopf approximation Expansion in powers of G F Use rest-frame of the meson [M i2 ] Γ = M M δ (0) ij + 1 2M M ij Sum includes phase-space of final state ˆ= Number of H weak Insertions n M i H weak n n H weak M j E +... n + iɛ M (0) M Decompose into dispersive and absorpative part optical theorem ( ) 1 1 ω + iɛ = P iπδ(ω) ω Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 10 / 30

17 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters The Hamilton Matrix: Computing Mixing Parameters Matrix to be understood in M 1 M M M 2 space Weak interaction sets scale: Wigner-Weisskopf approximation Expansion in powers of G F Use rest-frame of the meson [M i2 ] Γ = M M δ (0) ij + 1 2M M ij Sum includes phase-space of final state ˆ= Number of H weak Insertions n M i H weak n n H weak M j E +... n + iɛ M (0) M Decompose into dispersive and absorpative part optical theorem ( ) 1 1 ω + iɛ = P iπδ(ω) ω Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 10 / 30

18 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters The Hamilton Matrix: Computing Mixing Parameters Matrix to be understood in M 1 M M M 2 space Weak interaction sets scale: Wigner-Weisskopf approximation Expansion in powers of G F Use rest-frame of the meson [M i2 ] Γ = M M δ (0) ij + 1 2M M ij Sum includes phase-space of final state ˆ= Number of H weak Insertions n M i H weak n n H weak M j E +... n + iɛ M (0) M Decompose into dispersive and absorpative part optical theorem ( ) 1 1 ω + iɛ = P iπδ(ω) ω Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 10 / 30

19 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of Γ 12 Absorpative Part Γ ij = 1 M i H weak n n H weak M j (2π)δ(M (0) M 2M E n) M n On-shell production of intermediate particles i = j recovers total width dominated by B = 1 operator u, c b Only u and c intermediate state quarks b Calculation Perturbative corrections 1 Up to NLO in α s (m b ) 2 All-order summation of α n s (m 2 c/m 2 b )n log(m 2 c/m 2 b ) s u, c s [arxiv: ] Non-perturbative corrections up to Λ QCD /m b (5 more operators) Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 11 / 30

20 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of Γ 12 Absorpative Part Γ ij = 1 M i H weak n n H weak M j (2π)δ(M (0) M 2M E n) M n On-shell production of intermediate particles i = j recovers total width dominated by B = 1 operator u, c b Only u and c intermediate state quarks b Calculation Perturbative corrections 1 Up to NLO in α s (m b ) 2 All-order summation of α n s (m 2 c/m 2 b )n log(m 2 c/m 2 b ) s u, c s [arxiv: ] Non-perturbative corrections up to Λ QCD /m b (5 more operators) Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 11 / 30

21 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of Γ 12 Absorpative Part Γ ij = 1 M i H weak n n H weak M j (2π)δ(M (0) M 2M E n) M n On-shell production of intermediate particles i = j recovers total width dominated by B = 1 operator u, c b Only u and c intermediate state quarks b Calculation Perturbative corrections 1 Up to NLO in α s (m b ) 2 All-order summation of α n s (m 2 c/m 2 b )n log(m 2 c/m 2 b ) s u, c s [arxiv: ] Non-perturbative corrections up to Λ QCD /m b (5 more operators) Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 11 / 30

22 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of M 12 Dispersive Part [hep-ph/ ] M ij = 1 M i H F =2 M j + 1 2M M 2M M dominated by B = 2 operator top quark dominate intermediate state n P M i H F =1 n n H F =1 M j E n M (0) M b s s b Result within SM M 12 = G 2 F M B 12π 2 M2 W (V tb V ts) 2 ˆη B S 0 ( m 2 t M 2 W ) f 2 B s B [hep-ph/ ] Lattice determination of Bag parameter B and decay constant f Bs Mass difference measured precisly Fixes M 12 for M 12 Γ 12 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 12 / 30

23 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of M 12 Dispersive Part [hep-ph/ ] M ij = 1 M i H F =2 M j + 1 2M M 2M M dominated by B = 2 operator top quark dominate intermediate state n P M i H F =1 n n H F =1 M j E n M (0) M b s s b Result within SM M 12 = G 2 F M B 12π 2 M2 W (V tb V ts) 2 ˆη B S 0 ( m 2 t M 2 W ) f 2 B s B [hep-ph/ ] Lattice determination of Bag parameter B and decay constant f Bs Mass difference measured precisly Fixes M 12 for M 12 Γ 12 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 12 / 30

24 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of A sl [hep-ph/ ] In the Standard Model Naming scheme for B s,d : A d,s SL Sum of both including production asymmetry: A b SL SM phase orginates from CKM mechanism (convention dependent) A d,s d,s SL Im Γ12 /Md,s 12 Γ d,s tan φ d,s M d,s Highly suppressed: Γ 12 /M 12 = O(mb 2/M2 W, m2 c/mb 2) Beyond the Standard Model Additional phases can be introduced in M 12 due to New Physics Introduces sensitivity to Re (Γ 12 /M 12 ) SM Enhanced sensitivity for BSM physics in this observable Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 13 / 30

25 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Calculation of A sl [hep-ph/ ] In the Standard Model Naming scheme for B s,d : A d,s SL Sum of both including production asymmetry: A b SL SM phase orginates from CKM mechanism (convention dependent) A d,s d,s SL Im Γ12 /Md,s 12 Γ d,s tan φ d,s M d,s Highly suppressed: Γ 12 /M 12 = O(mb 2/M2 W, m2 c/mb 2) Beyond the Standard Model Additional phases can be introduced in M 12 due to New Physics Introduces sensitivity to Re (Γ 12 /M 12 ) SM Enhanced sensitivity for BSM physics in this observable Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 13 / 30

26 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Standard Model Predictions [A.Lenz,U.Nierste: ,hep-ph/ ] Predictions for B s System 2 Γ12 s = (0.087 ± 0.021) ps 1 M s = (17.3 ± 2.6) ps 1 φ s = (0.22 ± 0.06) a s SL = (1.9 ± 0.3)10 5 Predictions for B d System 2 Γ d 12 = (2.74 ± 0.51) 10 3 ps 1 φ d = ( 4.3 ± 1.4) a d SL = (4.1 ± 0.6)10 4 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 14 / 30

27 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters Standard Model Predictions [A.Lenz,U.Nierste: ,hep-ph/ ] Predictions for B s System 2 Γ12 s = (0.087 ± 0.021) ps 1 M s = (17.3 ± 2.6) ps 1 φ s = (0.22 ± 0.06) a s SL = (1.9 ± 0.3)10 5 Predictions for B d System 2 Γ d 12 = (2.74 ± 0.51) 10 3 ps 1 φ d = ( 4.3 ± 1.4) a d SL = (4.1 ± 0.6)10 4 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 14 / 30

28 Current Experimental Situation Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters DØ Like-sign Di-muon measurement [ ] A b SL = [7.87 ± 1.72 (stat) ± 0.93 (syst)] 10 3 = (0.594 ± 0.022) A d SL + (0.406 ± 0.022) A s SL M s = ( ± 0.043) ps 1 [hep-ex/ ,lhcb-conf (005)] M d = (0.507 ± 0.004) ps 1 Heavy Flavor Averaging Group (HFAG) Γ s = (0.116 ± 0.019) ps 1 Γ s = (0.068 ± 0.027) ps 1 LHCb CDF Γ s = ( ) ps 1 DØ LHCb measurement of time dependent CP asymmetry φ s = ± 0.101(stat) ± 0.027(syst) rad New measurement tend to agree well with SM [LHCb-CONF ] Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 15 / 30

29 Current Experimental Situation Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters DØ Like-sign Di-muon measurement [ ] A b SL = [7.87 ± 1.72 (stat) ± 0.93 (syst)] 10 3 = (0.594 ± 0.022) A d SL + (0.406 ± 0.022) A s SL M s = ( ± 0.043) ps 1 [hep-ex/ ,lhcb-conf (005)] M d = (0.507 ± 0.004) ps 1 Heavy Flavor Averaging Group (HFAG) Γ s = (0.116 ± 0.019) ps 1 Γ s = (0.068 ± 0.027) ps 1 LHCb CDF Γ s = ( ) ps 1 DØ LHCb measurement of time dependent CP asymmetry φ s = ± 0.101(stat) ± 0.027(syst) rad New measurement tend to agree well with SM [LHCb-CONF ] Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 15 / 30

30 Current Experimental Situation Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters DØ Like-sign Di-muon measurement [ ] A b SL = [7.87 ± 1.72 (stat) ± 0.93 (syst)] 10 3 = (0.594 ± 0.022) A d SL + (0.406 ± 0.022) A s SL M s = ( ± 0.043) ps 1 [hep-ex/ ,lhcb-conf (005)] M d = (0.507 ± 0.004) ps 1 Heavy Flavor Averaging Group (HFAG) Γ s = (0.116 ± 0.019) ps 1 Γ s = (0.068 ± 0.027) ps 1 LHCb CDF Γ s = ( ) ps 1 DØ LHCb measurement of time dependent CP asymmetry φ s = ± 0.101(stat) ± 0.027(syst) rad New measurement tend to agree well with SM [LHCb-CONF ] Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 15 / 30

31 Motivation Description of Meson Oscillation Theoretical Predictions of Oscillation Parameters How Can New Physics Enter? Can induce new operator structures Can modify Wilson coefficients Parametrization and Effects For B mesons: Γ 12 /M 12 1 m = 2M 12, Γ = 2 Γ 12 cos φ 12, A sl = 2δ = Im (Γ 12 /M 12 ) In general: Changing phase and absolute value possible M 12 = M SM 12 s e iφ s 1 Only relative phase relevant 2 M 12 : Heavy intermediate particles, but M 12 constrained by exp. 3 Γ 12 dominated by on-shell charm intermediate states Believed not to change dramatically by New Physics contribtions Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 16 / 30

32 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Generic Conditions on Mixing Parameter Physical Constraints Mass and width of physical states have to be positive Unitarity has to be conserved Time evolution of any linear combination of B 0 and B 0 determined entirely by the Γ matrix Γ itself has positive eigenvalues Defining Γ = (Γ H + Γ L )/2, x = (m H m L )/Γ and y = (Γ L Γ H )/(2Γ ) δ 2 < Γ H Γ L (m H m L ) 2 + (Γ H + Γ L ) 2 /4 = 1 y x 2 Known as unitarity bound or Bell-Steinberger inequality [J.S. Bell, J. Steinberger, Weak interactions of kaons, in R. G. Moorhouse et al., Eds., Proceedings of the Oxford Int. Conf. on Elementary Particles, Rutherford Laboratory, Chilton, England, 1965, p. 195.] Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 17 / 30

33 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Generic Conditions on Mixing Parameter Physical Constraints Mass and width of physical states have to be positive Unitarity has to be conserved Time evolution of any linear combination of B 0 and B 0 determined entirely by the Γ matrix Γ itself has positive eigenvalues Defining Γ = (Γ H + Γ L )/2, x = (m H m L )/Γ and y = (Γ L Γ H )/(2Γ ) δ 2 < Γ H Γ L (m H m L ) 2 + (Γ H + Γ L ) 2 /4 = 1 y x 2 Known as unitarity bound or Bell-Steinberger inequality [J.S. Bell, J. Steinberger, Weak interactions of kaons, in R. G. Moorhouse et al., Eds., Proceedings of the Oxford Int. Conf. on Elementary Particles, Rutherford Laboratory, Chilton, England, 1965, p. 195.] Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 17 / 30

34 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Generic Conditions on Mixing Parameter Physical Constraints Mass and width of physical states have to be positive Unitarity has to be conserved Time evolution of any linear combination of B 0 and B 0 determined entirely by the Γ matrix Γ itself has positive eigenvalues Defining Γ = (Γ H + Γ L )/2, x = (m H m L )/Γ and y = (Γ L Γ H )/(2Γ ) δ 2 < Γ H Γ L (m H m L ) 2 + (Γ H + Γ L ) 2 /4 = 1 y x 2 Known as unitarity bound or Bell-Steinberger inequality [J.S. Bell, J. Steinberger, Weak interactions of kaons, in R. G. Moorhouse et al., Eds., Proceedings of the Oxford Int. Conf. on Elementary Particles, Rutherford Laboratory, Chilton, England, 1965, p. 195.] Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 17 / 30

35 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Reminder: Sketching Derivation of Unitarity Bound Definitions a i = 2πρ i f i H B, ā i = 2πρ i f i H B a i a i = Γ 11, ā i ā i = Γ 22, ā i a i = Γ 12 In the physical basis we have a i = 1 2p (a Hi + a Li ), ā i = 1 2q (a Li a Hi ) ahi a Hi = Γ H, ali a Li = Γ L, ahi a Li = i(m H m L + iγ ) δ Sketch of Derivation Apply optical theorem and Cauchy Schwartz inequality Forces Γ to have positive semi-definit eigenvalues Γ 11 Γ 12 Apply the same to the physical basis with unitarity condition Leads to the unitarity bound δ 2 < Γ H Γ L (m H m L ) 2 +(Γ H +Γ L ) 2 /4 = 1 y 2 1+x 2 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 18 / 30

36 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Reminder: Sketching Derivation of Unitarity Bound Definitions a i = 2πρ i f i H B, ā i = 2πρ i f i H B a i a i = Γ 11, ā i ā i = Γ 22, ā i a i = Γ 12 In the physical basis we have a i = 1 2p (a Hi + a Li ), ā i = 1 2q (a Li a Hi ) ahi a Hi = Γ H, ali a Li = Γ L, ahi a Li = i(m H m L + iγ ) δ Sketch of Derivation Apply optical theorem and Cauchy Schwartz inequality Forces Γ to have positive semi-definit eigenvalues Γ 11 Γ 12 Apply the same to the physical basis with unitarity condition Leads to the unitarity bound δ 2 < Γ H Γ L (m H m L ) 2 +(Γ H +Γ L ) 2 /4 = 1 y 2 1+x 2 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 18 / 30

37 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Reminder: Sketching Derivation of Unitarity Bound Definitions a i = 2πρ i f i H B, ā i = 2πρ i f i H B a i a i = Γ 11, ā i ā i = Γ 22, ā i a i = Γ 12 In the physical basis we have a i = 1 2p (a Hi + a Li ), ā i = 1 2q (a Li a Hi ) ahi a Hi = Γ H, ali a Li = Γ L, ahi a Li = i(m H m L + iγ ) δ Sketch of Derivation Apply optical theorem and Cauchy Schwartz inequality Forces Γ to have positive semi-definit eigenvalues Γ 11 Γ 12 Apply the same to the physical basis with unitarity condition Leads to the unitarity bound δ 2 < Γ H Γ L (m H m L ) 2 +(Γ H +Γ L ) 2 /4 = 1 y 2 1+x 2 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 18 / 30

38 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Reminder: Sketching Derivation of Unitarity Bound Definitions a i = 2πρ i f i H B, ā i = 2πρ i f i H B a i a i = Γ 11, ā i ā i = Γ 22, ā i a i = Γ 12 In the physical basis we have a i = 1 2p (a Hi + a Li ), ā i = 1 2q (a Li a Hi ) ahi a Hi = Γ H, ali a Li = Γ L, ahi a Li = i(m H m L + iγ ) δ Sketch of Derivation Apply optical theorem and Cauchy Schwartz inequality Forces Γ to have positive semi-definit eigenvalues Γ 11 Γ 12 Apply the same to the physical basis with unitarity condition Leads to the unitarity bound δ 2 < Γ H Γ L (m H m L ) 2 +(Γ H +Γ L ) 2 /4 = 1 y 2 1+x 2 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 18 / 30

39 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Reminder: Sketching Derivation of Unitarity Bound Definitions a i = 2πρ i f i H B, ā i = 2πρ i f i H B a i a i = Γ 11, ā i ā i = Γ 22, ā i a i = Γ 12 In the physical basis we have a i = 1 2p (a Hi + a Li ), ā i = 1 2q (a Li a Hi ) ahi a Hi = Γ H, ali a Li = Γ L, ahi a Li = i(m H m L + iγ ) δ Sketch of Derivation Apply optical theorem and Cauchy Schwartz inequality Forces Γ to have positive semi-definit eigenvalues Γ 11 Γ 12 Apply the same to the physical basis with unitarity condition Leads to the unitarity bound δ 2 < Γ H Γ L (m H m L ) 2 +(Γ H +Γ L ) 2 /4 = 1 y 2 1+x 2 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 18 / 30

40 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Implication for the Three Neutral Mesons Neutral K Mesons Very limited amount of final states Unitarity bound developed for this case Neutral D Mesons Difficult in theory as well as in experiment 1 Non-perturbative methods? 2 Huge GIM suppression Interesting because of only up-type Neutral B d,s Mesons Lots of possible final states Experimental precision is increasing Systematic expansion for theoretical calculations possible Good opportunity to search for New Physics Improve bound on parameters Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 19 / 30

41 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Implication for the Three Neutral Mesons Neutral K Mesons Very limited amount of final states Unitarity bound developed for this case Neutral D Mesons Difficult in theory as well as in experiment 1 Non-perturbative methods? 2 Huge GIM suppression Interesting because of only up-type Neutral B d,s Mesons Lots of possible final states Experimental precision is increasing Systematic expansion for theoretical calculations possible Good opportunity to search for New Physics Improve bound on parameters Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 19 / 30

42 Improvement: Using Theoretical Input Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Assumptions Assume knowledge of Γ 12 Can be computed in a reliable, systematical expansion in the B system Define y 12 0 with 0 y 12 = Γ 12 / Γ 1 Goals Use precise measurement of mass difference Use a reliable theory prediction Obtain a relation between two measurable quantities Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 20 / 30

43 Improvement: Using Theoretical Input Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Assumptions Assume knowledge of Γ 12 Can be computed in a reliable, systematical expansion in the B system Define y 12 0 with 0 y 12 = Γ 12 / Γ 1 Goals Use precise measurement of mass difference Use a reliable theory prediction Obtain a relation between two measurable quantities Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 20 / 30

44 Improvement: Using Theoretical Input Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Assumptions Assume knowledge of Γ 12 Can be computed in a reliable, systematical expansion in the B system Define y 12 0 with 0 y 12 = Γ 12 / Γ 1 Goals Use precise measurement of mass difference Use a reliable theory prediction Obtain a relation between two measurable quantities Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 20 / 30

45 Deriving the Relation Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Sketch of the Steps Start with the same equations as for the unitarity bound Use y 12 instead of unitarity constraint Proceed with same steps The Result δ 2 = y 12 2 y 2 y x 2 = Γ 12 2 ( Γ ) 2 /4 Γ ( m) 2 Entirely determined by solving the eigenstate problem Relation is exact Monotonic function in y 12 In view of uncertainties, should be seen as an upper bound Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 21 / 30

46 Deriving the Relation Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Sketch of the Steps Start with the same equations as for the unitarity bound Use y 12 instead of unitarity constraint Proceed with same steps The Result δ 2 = y 12 2 y 2 y x 2 = Γ 12 2 ( Γ ) 2 /4 Γ ( m) 2 Entirely determined by solving the eigenstate problem Relation is exact Monotonic function in y 12 In view of uncertainties, should be seen as an upper bound Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 21 / 30

47 Deriving the Relation Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Sketch of the Steps Start with the same equations as for the unitarity bound Use y 12 instead of unitarity constraint Proceed with same steps The Result δ 2 = y 12 2 y 2 y x 2 = Γ 12 2 ( Γ ) 2 /4 Γ ( m) 2 Entirely determined by solving the eigenstate problem Relation is exact Monotonic function in y 12 In view of uncertainties, should be seen as an upper bound Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 21 / 30

48 Scaling Argument and CPT Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Obtain a Physical Understanding Relation can also be obtained from a scaling argument δ depends only on mixing parameters and independent of Γ Scale Γ by y 12 1 Does not affect δ 2 Changes x x/y 12 and y y/y 12 Combining argument with unitarity bound recovers exact relation Derivation not Assuming CPT invariance Assuming no CPT invariance implies M 11 M 22 and Γ 11 Γ 22 Mixing parameters depend on difference of diagonal components Relation applies for δ 2 Usual derivation do not go through if CPT is violated Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 22 / 30

49 Scaling Argument and CPT Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Obtain a Physical Understanding Relation can also be obtained from a scaling argument δ depends only on mixing parameters and independent of Γ Scale Γ by y 12 1 Does not affect δ 2 Changes x x/y 12 and y y/y 12 Combining argument with unitarity bound recovers exact relation Derivation not Assuming CPT invariance Assuming no CPT invariance implies M 11 M 22 and Γ 11 Γ 22 Mixing parameters depend on difference of diagonal components Relation applies for δ 2 Usual derivation do not go through if CPT is violated Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 22 / 30

50 Scaling Argument and CPT Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Obtain a Physical Understanding Relation can also be obtained from a scaling argument δ depends only on mixing parameters and independent of Γ Scale Γ by y 12 1 Does not affect δ 2 Changes x x/y 12 and y y/y 12 Combining argument with unitarity bound recovers exact relation Derivation not Assuming CPT invariance Assuming no CPT invariance implies M 11 M 22 and Γ 11 Γ 22 Mixing parameters depend on difference of diagonal components Relation applies for δ 2 Usual derivation do not go through if CPT is violated Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 22 / 30

51 Scaling Argument and CPT Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Obtain a Physical Understanding Relation can also be obtained from a scaling argument δ depends only on mixing parameters and independent of Γ Scale Γ by y 12 1 Does not affect δ 2 Changes x x/y 12 and y y/y 12 Combining argument with unitarity bound recovers exact relation Derivation not Assuming CPT invariance Assuming no CPT invariance implies M 11 M 22 and Γ 11 Γ 22 Mixing parameters depend on difference of diagonal components Relation applies for δ 2 Usual derivation do not go through if CPT is violated Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 22 / 30

52 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Combined Bound on A b SL Experimental Situation Hadron colliders produce admixture of B s and B d Production asymmetry is known at DØ A b SL = (0.594 ± 0.022) A d SL + (0.406 ± 0.022) A s SL B factories can access B d need Super-B for sufficient precision Implication for Unitarity Relation with Theory Input Relation (bound) on δ Relation for individual A d,s SL With know production asymmetry we can give a bound on A b SL (1.188 ± 0.044) δ d max + (0.812 ± 0.044) δ s max Bound is the same, if difference is measured Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 23 / 30

53 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Combined Bound on A b SL Experimental Situation Hadron colliders produce admixture of B s and B d Production asymmetry is known at DØ A b SL = (0.594 ± 0.022) A d SL + (0.406 ± 0.022) A s SL B factories can access B d need Super-B for sufficient precision Implication for Unitarity Relation with Theory Input Relation (bound) on δ Relation for individual A d,s SL With know production asymmetry we can give a bound on A b SL (1.188 ± 0.044) δ d max + (0.812 ± 0.044) δ s max Bound is the same, if difference is measured Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 23 / 30

54 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Combined Bound on A b SL Experimental Situation Hadron colliders produce admixture of B s and B d Production asymmetry is known at DØ A b SL = (0.594 ± 0.022) A d SL + (0.406 ± 0.022) A s SL B factories can access B d need Super-B for sufficient precision Implication for Unitarity Relation with Theory Input Relation (bound) on δ Relation for individual A d,s SL With know production asymmetry we can give a bound on A b SL (1.188 ± 0.044) δ d max + (0.812 ± 0.044) δ s max Bound is the same, if difference is measured Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 23 / 30

55 Plot of the Bound Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data 10 3 A SL b Assuming Γ d = 0 Shaded regions are allowed by theory prediction 1 Darker uses 1σ upper range 2 Lighter uses 2σ upper range Horizontal lines: 1σ range of A b SL DØ measurement Vertical lines correspond to Γ s LHCb measurement Dashed [dotted] curves: Mixed sigma interval of theory predictions The vertical boundaries of the shaded regions arise because Γ s > 2 Γ12 s is unphysical. Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 24 / 30

56 Plot of the Bound Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data 10 3 A SL b Assuming Γ d = 0 Shaded regions are allowed by theory prediction 1 Darker uses 1σ upper range 2 Lighter uses 2σ upper range Horizontal lines: 1σ range of A b SL DØ measurement Vertical lines correspond to Γ s LHCb measurement Dashed [dotted] curves: Mixed sigma interval of theory predictions The vertical boundaries of the shaded regions arise because Γ s > 2 Γ12 s is unphysical. Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 24 / 30

57 Plot of the Bound Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data 10 3 A SL b Assuming Γ d = 0 Shaded regions are allowed by theory prediction 1 Darker uses 1σ upper range 2 Lighter uses 2σ upper range Horizontal lines: 1σ range of A b SL DØ measurement Vertical lines correspond to Γ s LHCb measurement Dashed [dotted] curves: Mixed sigma interval of theory predictions The vertical boundaries of the shaded regions arise because Γ s > 2 Γ12 s is unphysical. Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 24 / 30

58 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Plot on Individual Bound on B s 0.15 s ps Interpretation Horizontal lines correspond to LHCb measurement Dark [light] shaded allowed by 1σ [2σ] theory variation No discrepancy claimed in experiment Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 25 / 30

59 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Plot on Individual Bound on B d 10 3 d ps Interpretation Dark [light] shaded allowed by 1σ [2σ] theory variation No discrepancy claimed in experiment Non-zero measurement of Γ d would strengthen upper bound Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 26 / 30

60 Unitarity Constraint Deriving a Relation using Theoretical Input Application to Recent Data Plot on Individual Bound on B d 10 3 d ps Interpretation Dark [light] shaded allowed by 1σ [2σ] theory variation No discrepancy claimed in experiment Non-zero measurement of Γ d would strengthen upper bound Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 26 / 30

61 Numerical Interpretation Discussuion Remarks Problematic: Γ s meas. > 2 Γ12 s is unphysical Numerator of Relation can vanish Upper bound Assume 2σ theory prediction for a conservative estimate Results For B s system, we obtain by propagating the uncertainties, taking into account the unphysical region A s SL < times better than best current experimental bound For the B d system we obtain a comparable bound A d SL < Significant improvement possible by observing Γ d > 0 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 27 / 30

62 Numerical Interpretation Discussuion Remarks Problematic: Γ s meas. > 2 Γ12 s is unphysical Numerator of Relation can vanish Upper bound Assume 2σ theory prediction for a conservative estimate Results For B s system, we obtain by propagating the uncertainties, taking into account the unphysical region A s SL < times better than best current experimental bound For the B d system we obtain a comparable bound A d SL < Significant improvement possible by observing Γ d > 0 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 27 / 30

63 Numerical Interpretation Discussuion Remarks Problematic: Γ s meas. > 2 Γ12 s is unphysical Numerator of Relation can vanish Upper bound Assume 2σ theory prediction for a conservative estimate Results For B s system, we obtain by propagating the uncertainties, taking into account the unphysical region A s SL < times better than best current experimental bound For the B d system we obtain a comparable bound A d SL < Significant improvement possible by observing Γ d > 0 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 27 / 30

64 Discussion of the Results Strength of the Bound Discussuion Upper bound on y 12 implies an upper bound on δ Relation is much stronger for small y 12, as e.g. in the B d system Comparing to Known Results DØ A SL measurement: 3.9σ discrepancy with SM Correlated with the discrepancy found in our analysis 1 SM prediction of A SL uses calculation of Γ 12 2 The relation uses Γ 12 as an input 3 Calculation of Γ 12 and Im(Γ 12 ) rely on the same OPE Large cancellations in Im(Γ 12 ) Uncertainties could be larger than expected from NLO calculation [hep-ph/ , hep-ph/ ] The sensitivity of Γ 12 to NP is generally weak Interesting to determine δ additionally from this relation Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 28 / 30

65 Discussion of the Results Strength of the Bound Discussuion Upper bound on y 12 implies an upper bound on δ Relation is much stronger for small y 12, as e.g. in the B d system Comparing to Known Results DØ A SL measurement: 3.9σ discrepancy with SM Correlated with the discrepancy found in our analysis 1 SM prediction of A SL uses calculation of Γ 12 2 The relation uses Γ 12 as an input 3 Calculation of Γ 12 and Im(Γ 12 ) rely on the same OPE Large cancellations in Im(Γ 12 ) Uncertainties could be larger than expected from NLO calculation [hep-ph/ , hep-ph/ ] The sensitivity of Γ 12 to NP is generally weak Interesting to determine δ additionally from this relation Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 28 / 30

66 Discussion of the Results Strength of the Bound Discussuion Upper bound on y 12 implies an upper bound on δ Relation is much stronger for small y 12, as e.g. in the B d system Comparing to Known Results DØ A SL measurement: 3.9σ discrepancy with SM Correlated with the discrepancy found in our analysis 1 SM prediction of A SL uses calculation of Γ 12 2 The relation uses Γ 12 as an input 3 Calculation of Γ 12 and Im(Γ 12 ) rely on the same OPE Large cancellations in Im(Γ 12 ) Uncertainties could be larger than expected from NLO calculation [hep-ph/ , hep-ph/ ] The sensitivity of Γ 12 to NP is generally weak Interesting to determine δ additionally from this relation Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 28 / 30

67 No Go Theorem (preliminary) Discussuion The Claim There is no generic bound, stronger than the unitarity bound Sketch of Derivation Unitarity bound is satured if f T B H f T B L Start with an arbitrary, generic decaying two-state system Wigner-Weisskopf approximation: Any choice of parameters OK Orthogonal, non CP violating system as starting point Arbitrary new UV physics can change M 12 independently of Γ 12 Varying M 12 keeping mass and width of states physical Unitarity bound can be satured (relax constraint Arg M 12 = Arg Γ 12 ) Explicit mathematical check Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 29 / 30

68 No Go Theorem (preliminary) Discussuion The Claim There is no generic bound, stronger than the unitarity bound Sketch of Derivation Unitarity bound is satured if f T B H f T B L Start with an arbitrary, generic decaying two-state system Wigner-Weisskopf approximation: Any choice of parameters OK Orthogonal, non CP violating system as starting point Arbitrary new UV physics can change M 12 independently of Γ 12 Varying M 12 keeping mass and width of states physical Unitarity bound can be satured (relax constraint Arg M 12 = Arg Γ 12 ) Explicit mathematical check Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 29 / 30

69 No Go Theorem (preliminary) Discussuion The Claim There is no generic bound, stronger than the unitarity bound Sketch of Derivation Unitarity bound is satured if f T B H f T B L Start with an arbitrary, generic decaying two-state system Wigner-Weisskopf approximation: Any choice of parameters OK Orthogonal, non CP violating system as starting point Arbitrary new UV physics can change M 12 independently of Γ 12 Varying M 12 keeping mass and width of states physical Unitarity bound can be satured (relax constraint Arg M 12 = Arg Γ 12 ) Explicit mathematical check Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 29 / 30

70 No Go Theorem (preliminary) Discussuion The Claim There is no generic bound, stronger than the unitarity bound Sketch of Derivation Unitarity bound is satured if f T B H f T B L Start with an arbitrary, generic decaying two-state system Wigner-Weisskopf approximation: Any choice of parameters OK Orthogonal, non CP violating system as starting point Arbitrary new UV physics can change M 12 independently of Γ 12 Varying M 12 keeping mass and width of states physical Unitarity bound can be satured (relax constraint Arg M 12 = Arg Γ 12 ) Explicit mathematical check Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 29 / 30

71 No Go Theorem (preliminary) Discussuion The Claim There is no generic bound, stronger than the unitarity bound Sketch of Derivation Unitarity bound is satured if f T B H f T B L Start with an arbitrary, generic decaying two-state system Wigner-Weisskopf approximation: Any choice of parameters OK Orthogonal, non CP violating system as starting point Arbitrary new UV physics can change M 12 independently of Γ 12 Varying M 12 keeping mass and width of states physical Unitarity bound can be satured (relax constraint Arg M 12 = Arg Γ 12 ) Explicit mathematical check Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 29 / 30

72 Discussuion Provided a physical derivation of the exact relation allowing for theoretical input on Γ 12 1 Input is typically insensitive to New Physics 2 Avoids largest uncertainties of theory calculation 3 Valid even if CPT is violated Independent of the discrepancy found from a global fit 1 Application to B d,s systems leads to the individual bounds A s SL < A d SL < Providing a bound on the individual asymmetries at comparable or better levels than the current experimental bounds 3 Bounds are in tension with the DØ measurement of A b SL Once an unambiguous determination of A SL or Γ is made, we can use it to constrain the other observable. Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 30 / 30

73 Backup Slides Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 31 / 30

74 Direct CP Violation General Comment Need CP even and odd phases Γ A 1 (f ) + A 2 (f ) 2 [Bigi,Sanda: CP violation] Interference of CP conserving (strong) and violating (weak) phases Occurs in neutral and charged meson decays Neccessary condition: A(f ) Ā( f ) Example of Process Only source of CP violation in charged meson decays A f ± Γ (P f ) Γ (P + f + ) Γ (P f ) + Γ (P + f + ) = Ā(f )/A(f + ) 2 1 Ā(f )/A(f + ) Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 32 / 30

75 CP Violation in Interference of Mixing and Decay General Comments [Bigi,Sanda: CP violation] Interference between decay and mixing to common final state Neccessary condition: [ ] q Ā(f ) Im p A(f ) 0 Example of Process CP Asymmetry (easy form only in limits, e.g. B mesons) A fcp (t) Γ ( P 0 f CP ) Γ (P 0 f CP ) Γ ( P 0 f CP ) + Γ (P 0 f CP ) cos( Mt) Amix CP sin( Mt) cosh( Γ t/2) + A Γ sinh( Γ t/2) = Adir CP B s J/Ψφ and B s J/Ψf 0 Sascha Turczyk Constraining CP violation in neutral meson mixing with theory input 33 / 30