1. : Section 3: Mixing and CP violation in (mostly) neutral mesons

2. State evolution in neutral mesons Neutral meson states P o, P o –Could be K o, D o, B o With internal quantum number F –Such that  F=0 emag/strong,  F  0 for H weak Obeys time dependent Schrödinger eqn M,  Hermitian 2x2 matrices, mass matrix and decay matrix, H 11 =H 22 from CPT (mass/decay particle = anti- particle)

3. Solve Schrödinger for Eigenstates of H w From characteristic equation If E 1 =M 1 -i  1 /2, E 2 = M 2 -i  2 /2 and  M=M 2 -M 1,  =  2 -  1 Complex coeff. p,q obey |q| 2 +|p| 2 =1

4. Eigenvector Eqn Eigenstates have time evolution: then with

5. Finally probabilities… With The oscillations depend upon the parameter x the speed of oscillations c.f. lifetime Where  = (  1 +  2 )/2 Interference term

6. Mixing probabilities x in B s not yet measured >19 Probability of finding P o from initial pure P o beam

7. CP in decay CP in mixing CP in interference between mixing and decay Types of CP violation P f f P f f P P f f P PP P f f P PP P + +

8. 1) CP Violation in mixing Indirect CP Violation Mass eigenstates being different from CP eigenstates Mixing rate P o  P o is different to P o  P o If conserved CP|P 1 > = +1 |P 1 > CP|P 2 > = -1 |P 2 > with If violated Such asymmetries usually small Need to calculate M, , involve hadronic uncertainties Hence, tricky to relate to CKM parameters

9. 2) CP Violation in decay direct CP Violation Two types of phase –Weak phase:  due to weak interactions (complex CKM elements) –Strong phase: contribution from intermediate states, CP conserving (same sign in both) f is final state common to both decays occurs for both charged and neutral states P

10. 3)CP violation in the interference of mixing and decay Choose state* f, P o  f, P o  f Two possible decay chains, with or w/o mixing CP can be conserved in mixing and in decay and still be violated ! *Not necessary to be CP eigenstate Interference term depends on Can putand getbut

11. : Section 4: Neutral Kaon system

12. K o K o system Kaon mesons in two isospin doublets K + = us K o = ds K - = us K o = ds S=+1S=-1 Part of pseudo-scalar J P =0 - meson octet with ,  I 3 =+1/2 I 3 =-1/2 Kaon production K o :  - + p  o + K o But from baryon number conservation: K o :  + + p  K + + K o + p Or K o :  - + p  o + K o + n +n Requires higher energy Much higher S 0 0 -1 +1 S 0 0 +1 -1 0 S 0 0 +1 -1 0 0

13. Kaon oscillations So say at t=0, pure K o, –later a superposition of states d s u, c, t WW WW _ s d _ d s WW WW _ s d _ _ _ _ K0K0 K0K0 

14. K o Decay In that case CP=+1 CP=-1 K s  o  o K s  +  - K L  +  -  o K L  o  o  o CP=+1 CP=-1 K S branching fractions:      69%,      31% K L branching fractions:        21%,        13%,     66% mass eigenstates KSKS KLKL show Assume CP

15. Time dependent probabilities for the neutral kaon case. t (1/   )

16. K o Regeneration Start with Pure K o beam –After time all K s component decayed Introduce slab of material in beam –reactions 1) Elastic scatttering2) Charge exchange 3) Hyperon production Hence K o absorbed more strongly i.e. K s regenerated

17. Discovery of CP Violation K 1  o  o K 1  +  - K 2  +  -  o K 2  o  o  o CP=+1 CP=-1 So if K L =K 1 CP eigenstate, Observe no two pion component But if broken get: Where  quantifies degree of CP violation BUT Can one find K L decaying into  +  - ?

18. K L       X K L  p  = p  + p   = angle between p KL and p  If X = 0, p  = p KL  : cos  = 1 If X  0, p   p KL  : cos   1 cos  KLKL  m (     ) < m K J.H. Christenson et al., PRL 13,138 (1964) Discovery of CP violation m (     ) = m K m (     ) > m K Mass and angular spectrum

19. So CP symmetry is violated in the neutral kaon system. Mass eigenstates (K S and K L )  CP eigenstates Both K S and K L could decay into  + -  -. Experimentally well known: The majority of K S decays into   -   and K L into   -   -  . Small but with profound implications In K L

20. Decay final state at time t -- Spin(  ) = 0 L  -  = 0 Initial state at t = 0 S = 0 CP(   -    i.e. CP eigenstate K 0 at t = 0 decays into     vs K 0 at t = 0 decays into     any difference = CP violation CPLEAR revisited Tag K o /K o from charged pion/kaon

21. K0K0 K0K0 _ CPLEAR R +- (t) and R +- (t) _ CP violation

22. CPLEAR CP asymmetry A  (t) = R +- (t)  R +- (t) R +- (t)  R +- (t) _ _ large difference! CP violation in mixing The two mass eigenstates are not CP eigenstates

23. Kaons: CP violation in Decay CP violation first through existence of certain decay modes ~2.3x10 -3 If CP violation is only in mixing, i.e. independent of decay So, put channel independent term  and channel dependent  ’ Hence, by measuring only rates: get

24. So, two expts in the 80’s did it: NA31 (CERN) E731 (Fermilab) Ambiguous result! So, two expts did it again…….

25. NA48

26. KTeV

27. Measure     and     at the same time: NA31, NA48 K S is regenerated from K L : E731, KTeV No normalization is required, but efficiencies, acceptances etc. have to be corrected… Normalisation constants

28. Effort over 30 years! Note; 3 Re = |   /   |   i.e.  Re  0           Not easy to compare with SM theory

29. CKM parameters with CP conserving parameters |Vud| : nuclear Beta decay 0.9734  0.0008 |Vus| : semileptonic Kaon and hyperon decay (CCFR) 0.2196  0.0026 |Vcd| : Neutrino and anti-neutrino production of charm off valence d quarks 0.224  0.016 |Vcs| : W decays (LEP/me!) 0.996  0.013 |Vcb| : semileptonic inclusive and exclusive B decays (LEP/CLEO) 0.0412  0.0020 |Vub| end point spectrum in semileptonic B decays(LEP/CLEO) 0.0036  0.0007 B o mixing x d, + lattice gauge inputs |Vtb*Vtd| 0.0079  0.0015 Can use Unitarity constraints