1. 1Chris Parkes Part II CP Violation in the SM Chris Parkes

2. 2 Outline THEORETICAL CONCEPTS I.Introductory concepts Matter and antimatter Symmetries and conservation laws Discrete symmetries P, C and T II.CP Violation in the Standard Model Kaons and discovery of CP violation Mixing in neutral mesons Cabibbo theory and GIM mechanism The CKM matrix and the Unitarity Triangle Types of CP violation

3. Kaons and discovery of CP violation

4. 4Chris Parkes What about the product CP? ++ ++  ++  ++ Intrinsic spin P C    CP Initially CP appears to be preserved in weak interactions …! Weak interactions experimentally proven to:  Violate P : Wu et al. experiment, 1956  Violate C : Lederman et al., 1956 (just think about the pion decay below and non-existence of right-handed neutrinos)  But is C+P  CP symmetry conserved or violated?

5. 5Chris Parkes Kaon mesons: in two isospin doublets Part of pseudo-scalar J P =0 - mesons octet with ,  Introducing kaons K + = us K o = ds K - = us S=+1S=-1 I 3 =+1/2 I 3 =-1/2  Kaon production: (pion beam hitting a target) 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

6. 6Chris Parkes  What precisely is a K 0 meson?  Now we know the quark contents: K 0 =sd, K 0 =sd  First: what is the effect of C and P on the K 0 and K 0 particles?  (because l=0 q qbar pair)  effect of CP :  Bottom line: the flavour eigenstates K 0 and K 0 are not CP eigenstates Neutral kaons (1/2)

7. 7Chris Parkes  Nevertheless it is possible to construct CP eigenstates as linear combinations  Can always be done in quantum mechanics, to construct CP eigenstates  |K 1 > = 1/  2(|K 0 > + |K 0 >)  |K 2 > = 1/  2(|K 0 > - |K 0 >) Then:  CP |K 1 > = +1 |K 1 >  CP |K 2 > = -1 |K 2 >  Does it make sense to look at these linear combinations?  i.e. do these represent real particles?  Predictions were:  The K 1 must decay to 2 pions given CP conservation of the weak interactions  This 2 pion neutral kaon decay was the decay observed and therefore known  The same arguments predict that K 2 must decay to 3 pions  History tells us it made sense!  The K 2 = K L (“K-long”) was discovered in 1956 after being predicted  (difference between K 2 and K L to be discussed later) Neutral kaons (2/2)

8. 8Chris Parkes  How do you obtain a pure ‘beam’ of K 2 particles?  It turns out that you can do that through clever use of kinematics  Exploit that decay of neutral K (K 1 ) into two pions is much faster than decay of neutral K (K 2 ) into three pions  Mass K 0 =498 MeV, Mass π 0, π +/- =135 / 140 MeV  Therefore K 2 must have a longer lifetime thank K 1 since small decay phase space   1 = ~0.9 x 10 -10 sec   2 = ~5.2 x 10 -8 sec (~600 times larger!)  Beam of neutral kaons automatically becomes beam of |K 2 > as all |K 1 > decay very early on… Looking closer at K L decays Initial K 0 beam K 1 decay early (into ) Pure K 2 beam after a while! (all decaying into πππ ) !

9. 9Chris Parkes Incoming K 2 beam Decay of K 2 into 3 pions If you detect two of the three pions of a K 2   decay they will generally not point along the beam line Essential idea: Look for (CP violating) K 2   decays 20 meters away from K 0 production point The Cronin & Fitch experiment (1/3) J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turley PRL 13,138 (1964) π0π0 π+π+ π-π- Vector sum of p(π - ),p(π + )

10. 10Chris Parkes Incoming K 2 beam Decaying pions If K 2 decays into two pions instead of three both the reconstructed direction should be exactly along the beamline (conservation of momentum in K 2   decay) The Cronin & Fitch experiment (2/3) J.H. Christenson et al., PRL 13,138 (1964) Essential idea: Look for (CP violating) K 2   decays 20 meters away from K 0 production point

11. 11Chris Parkes Result: an excess of events at =0 degrees! K 2   decays (CP Violation!) K 2   decays Note scale: 99.99% of K   decays are left of plot boundary The Cronin & Fitch experiment (3/3) K 2       X p  = p  + p   = angle between p K2 and p  If X = 0, p  = p K2  : cos  = 1 If X  0, p   p K2  : cos   1 Weak interactions violate CP Effect is tiny, ~0.05% ! Weak interactions violate CP Effect is tiny, ~0.05% !

12. 12Chris Parkes

13. 13Chris Parkes Almost but not quite!

14. 14Chris Parkes with |ε| <<1

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16. 16Chris Parkes Key Points So Far K 0, K 0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states CP Violated (a tiny bit) in Kaon decays Describe this through K s, K L as mixture of K 0 K 0

17. Mixing in neutral mesons HEALTH WARNING : We are about to change notation P 1,P 2 are like K s, K L (rather than K 1,K 2 )

18. 18Chris Parkes Particle can transform into its own anti-particle  neutral meson states P o, P o  P could be K o, D o, B o, or B s o 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  _

19. 19Chris Parkes

20. 20Chris Parkes Here for general derivation we have labelled states 1,2

21. 21Chris Parkes  neutral meson states P o, P o  P could be K o, D o, B o, or B s o  with internal quantum number F  Such that  F=0 strong/EM interactions but  F  0 for weak interactions  obeys time-dependent Schrödinger equation  M,  : hermitian 2x2 matrices, mass matrix and decay matrix  mass/lifetime particle = antiparticle  Solution of form No Mixing – Simplest Case

22. 22Chris Parkes  neutral meson states P o, P o  P could be K o, D o, B o, or B s o  with internal quantum number F  Such that  F=0 strong/EM interactions but  F  0 for weak interactions  obeys time-dependent Schrödinger equation  M,  : hermitian 2x2 matrices, mass matrix and decay matrix  H 11 =H 22 from CPT invariance (mass/lifetime particle = antiparticle) Time evolution of neutral mesons mixed states (1/4) H is the total hamiltonian: EM+strong+weak

23. 23Chris Parkes  Solve Schrödinger for the eigenstates of H :  of the form  with complex parameters p and q satisfying  Time evolution of the eigenstates: Time evolution of neutral mesons mixed states (2/4) Compare with K s, K L as mixtures of K 0, K 0 If equal mixtures, like K 1 K 2

24. 24Chris Parkes  Some facts and definitions:  Characteristic equation  Eigenvector equation: Time evolution of neutral mesons mixed states (3/4) e.g.

25. 25Chris Parkes  Evolution of weak/flavour eigenstates:  Time evolution of mixing probabilities: Interference term Time evolution of neutral mesons mixed states (4/4) i.e. if start with P 0, what is probability that after time t that have state P 0 ? decay terms Parameter x determines “speed” of oscillations compared to the lifetime

26. 26Chris Parkes Hints: for proving probabilities Starting point Turn this around, gives Time evolution Use these to find

27. 27Chris Parkes Δm d = 0.507 ± 0.004 ps −1 x d = 0.770 ± 0.008 Δm s = 17.719 ± 0.043 ps −1 x s = 26.63 ± 0.18 x = 0.00419 ± 0.00211 Lifetimes very different (factor 600)

28. 28Chris Parkes Key Points So Far K 0, K 0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states CP Violated (a tiny bit) in Kaon decays Describe this through K s, K L as mixture of K 0 K 0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P 0 P 0 Mass differences and width determine the rates of oscillations Very different for different mesons (B s,B,D,K)

29. 29Chris Parkes Cabibbo theory and GIM mechanism

30. 30Chris Parkes  In 1963 N. Cabibbo made the first step to formally incorporate strangeness violation in weak decays 1) For the leptons, transitions only occur within a generation 2) For quarks the amount of strangeness violation can be neatly described in terms of a rotation, where  c =13.1 o Cabibbo rotation and angle (1/3) Weak force transitions u d’ = dcos c + ssin c W+W+ Idea: weak interaction couples to different eigenstates than strong interaction weak eigenstates can be written as rotation of strong eigenstates

31. 31Chris Parkes  Cabibbo’s theory successfully correlated many decay rates by counting the number of cos  c and sin  c terms in their decay diagram: Cabibbo rotation and angle (2/3) E.g.

32. 32Chris Parkes  There was however one major exception which Cabibbo could not describe: K 0   +  - (branching ratio ~7.10 -9 ) – Observed rate much lower than expected from Cabibbo’s rate correlations (expected rate  g 8 sin 2  c cos 2  c ) Cabibbo rotation and angle (3/3) d ++ --  u cos c sin c W W s

33. 33Chris Parkes The GIM mechanism (1/2)  In 1970 Glashow, Iliopoulos and Maiani publish a model for weak interactions with a lepton-hadron symmetry  The weak interaction couples to a rotated set of down-type quarks: the up-type quarks weakly decay to “rotated” down-type quarks  The Cabibbo-GIM model postulates the existence of a 4 th quark : the charm (c) quark ! … discovered experimentally in 1974: J/  cc state Lepton sector unmixed Quark section mixed through rotation of weak w.r.t. strong eigenstates by  c 2D rotation matrix

34. 34Chris Parkes The GIM mechanism (2/2)  There is also an interesting symmetry between quark generations: u d’=cos( c )d+sin( c )s W+W+ c s’=-sin( c )d+cos( c )s W+W+ Cabibbo mixing matrix The d quark as seen by the W, the weak eigenstate d’, is not the same as the mass eigenstate (the d)

35. 35Chris Parkes GIM suppression d ++ --  u cos c sin c W W s d ++ --  c cos c -sin c W W s expected rate  (g 4 sin  c cos  c - g 4 sin  c cos  c ) 2 The cancellation is not perfect – these are only the vertex factors – as the masses of c and u are different See also B s   +  - discussion later  The model also explains the smallness of the K 0   +  - decay

36. The CKM matrix and the Unitarity Triangle

37. 37Chris Parkes How to incorporate CP violation in the SM? hence “anti-unitary” T (and CP) operation corresponds to complex conjugation ! Simple exercise: Since H = H(V ij ), complex V ij would generate [T,H]  0  CP violation = only if:  How does CP conjugation (or, equivalently, T conjugation) act on the Hamiltonian H ? CP conservation is: (up to unphysical phase) Recall:

38. 38Chris Parkes Brilliant idea from Kobayashi and Maskawa (Prog. Theor. Phys. 49, 652(1973) )  Try and extend number of families (based on GIM ideas). E.g. with 3: … as mass and flavour eigenstates need not be the same (  rotated)  In other words this matrix relates the weak states to the physical states u d’ c s’ t b’ The CKM matrix (1/2) Kobayashi Maskawa Imagine a new doublet of quarks 2D rotation matrix 3D rotation matrix

39. 39Chris Parkes  Standard Model weak charged current Feynman diagram amplitude proportional to V ij U i D j U (D) are up (down) type quark vectors  V ij is the quark mixing matrix, the CKM matrix for 3 families this is a 3x3 matrix U = uctuct D = dsbdsb The CKM matrix (2/2) Can estimate relative probabilities of transitions from factors of |V ij | 2

40. 40Chris Parkes  As the CKM matrix elements are connected to probabilities of transition, the matrix has to be unitary: CKM matrix – number of parameters (1/2) Values of elements: a purely experimental matter In general, for N generations, N 2 constraints Sum of probabilities must add to 1 e.g. t must decay to either b, s, or d so  Freedom to change phase of quark fields 2N-1 phases are irrelevant (choose i and j, i≠j)  Rotation matrix has N(N-1)/2 angles

41. 41Chris Parkes CKM matrix – number of parameters (2/2) Example for N = 1 generation: 2 unknowns – modulus and phase: unitarity determines |V | = 1 the phase is arbitrary (non-physical) no phase, no CPV  NxN complex element matrix: 2N 2 parameters Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases) Number of phases

42. 42Chris Parkes CKM matrix – number of parameters (2/2)  NxN complex element matrix: 2N 2 parameters Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases) Number of phases Example for N = 2 generations: 8 unknowns – 4 moduli and 4 phases unitarity gives 4 constraints : for 4 quarks, we can adjust 3 relative phases only one parameter, a rotation (= Cabibbo angle) left: no phase  no CPV

43. 43Chris Parkes CKM matrix – number of parameters (2/2)  NxN complex element matrix: 2N 2 parameters Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases) Number of phases Example for N = 3 generations: 18 unknowns – 9 moduli and 9 phases unitarity gives 9 constraints for 6 quarks, we can adjust 5 relative phases 4 unknown parameters left: 3 rotation (Euler) angles and 1 phase  CPV ! In requiring CP violation with this structure of weak interactions K&M predicted a 3 rd family of quarks!

44. 44Chris Parkes C 12 S 12 0 -S 12 C 12 0 0 0 1 10 0 0 C 23 S 23 0 -S 23 C 23  3 angles  12,  23,  13 phase  V CKM = R 23 x R 13 x R 12 R 12 = R 23 = R 13 = C 13 0 S 13 e -i  0 1 0 -S 13 e -i  0 C 13 CKM matrix – Particle Data Group (PDG) parameterization Define: C ij = cos  ij S ij =sin  ij 3D rotation matrix form

45. 45Chris Parkes A ~ 1,  ~ 0.22,  ≠ 0 but  ≠ 0 ??? Introduced in 1983:  3 angles = S 12, A = S 23 /S 2 12,  = S 13 cos  / S 13 S 23  1 phase  = S 13 sin  / S 12 S 23 V CKM (3) terms in up to 3  CKM terms in 4, 5 CKM matrix - Wolfenstein parameters Note: smallest couplings are complex (  CP-violation)

46. 46Chris Parkes A ~ 1,  ~ 0.22,  ≠ 0 but  ≠ 0 ??? Introduced in 1983:  3 angles = S 12, A = S 23 /S 2 12,  = S 13 cos  / S 13 S 23  1 phase  = S 13 sin  / S 12 S 23 V CKM (3) terms in up to 3  CKM terms in 4, 5 CKM matrix - Wolfenstein parameters Note: smallest couplings are complex (  CP-violation)

47. 47Chris Parkes A ~ 1,  ~ 0.22,  ≠ 0 but  ≠ 0 ??? Introduced in 1983:  3 angles = S 12, A = S 23 /S 2 12,  = S 13 cos  / S 13 S 23  1 phase  = S 13 sin  / S 12 S 23 V CKM (3) terms in up to 3  CKM terms in 4, 5 CKM matrix - Wolfenstein parameters Note: smallest couplings are complex (  CP-violation)

48. 48Chris Parkes CKM matrix - hierarchy  ~ 0.22 top charm up down strange bottom Charge: +2/3 Charge:  1/3 flavour-changing transitions by weak charged current (boldness indicates transition probability  |V ij |)

49. 49 CKM – Unitarity Triangle Three complex numbers, which sum to zero Divide by so that the middle element is 1 (and real) Plot as vectors on an Argand diagram If all numbers real – triangle has no area – No CP violation Real Imaginary Hence, get a triangle ‘Unitarity’ or ‘CKM triangle’ Triangle if SM is correct. Otherwise triangle will not close, Angles won’t add to 180 o

50. 50Chris Parkes Plot on Argand diagram: 6 triangles in complex plane : no phase info. db: sb: ds: ut: ct: uc: Unitarity conditions and triangles

51. 51Chris Parkes The Unitarity Triangle(s) & the , ,  angles  Area of all the triangles is the same ( 6 A 2  )  Jarlskog invariant J, related to how much CP violation  Two triangles (db) and (ut) have sides of similar size Easier to measure, (db) is often called THE unitarity triangle

52. 52Chris Parkes CKM Triangle - Experiment Find particle decays that are sensitive to measuring the angles (phase difference) and sides (probabilities) of the triangles Measurements constrain the apex of the triangle Measurements are consistent We will discuss how to experimentally measure the sides / angles CKM model works, 2008 Nobel prize

53. 53Chris Parkes Key Points So Far K 0, K 0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states CP Violated (a tiny bit) in Kaon decays Describe this through K s, K L as mixture of K 0 K 0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P 0 P 0 Mass differences and width determine the rates of oscillations Very different for different mesons (B s,B,D,K) Weak and mass eigenstates of quarks are not the same Describe through rotation matrix – Cabibbo (2 generations), CKM (3 generations) CP Violation included by making CKM matrix elements complex Depict matrix elements and their relationships graphically with CKM triangle

54. Types of CP violation We discussed earlier how CP violation can occur in Kaon (or any P 0 ) mixing if p≠q. We didn’t consider the decay of the particle – this leads to two more ways to violate CP

55. 55Chris Parkes  CP in decay  CP in mixing  CP in interference between mixing and decay P f f P f f P P f f P PP P f f P PP P + + Types of CP violation

56. 56Chris Parkes Occurs when a decay and its CP-conjugate decay have a different probability Decay amplitudes can be written as: Two types of phase:  Strong phase: CP conserving, contribution from intermediate states  Weak phase  : complex phase due to weak interactions 1) CP violation in decay (also called direct CP violation) Valid for both charged and neutral particles P (other types are neutral only since involve oscillations)

57. 57Chris Parkes  Mass eigenstates being different from CP eigenstates  Mixing rate for P 0  P 0 can be different from P 0  P 0  If CP conserved :  If CP violated : with  such asymmetries usually small  need to calculate M, , involve hadronic uncertainties  hence tricky to relate to CKM parameters 2) CP violation in mixing (also called indirect CP violation) (This is the case if K s =K 1, K L =K 2 )

58. 58Chris Parkes  Say we have a particle* such that P 0  f and P 0  f are both possible  There are then 2 possible decay chains, with or without mixing!  Interference term depends on  Can put and get but * Not necessary to be CP eigenstate 3) CP violation in the interference of mixing and decay CP can be conserved in mixing and in decay, and still be violated overall !

59. 59Chris Parkes Key Points So Far K 0, K 0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states CP Violated (a tiny bit) in Kaon decays Describe this through K s, K L as mixture of K 0 K 0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P 0 P 0 Mass differences and width determine the rates of oscillations Very different for different mesons (B s,B,D,K) Weak and mass eigenstates of quarks are not the same Describe through rotation matrix – Cabibbo (2 generations), CKM (3 generations) CP Violation included by making CKM matrix elements complex Depict matrix elements and their relationships graphically with CKM triangle Three ways for CP violation to occur Decay Mixing Interference between decay and mixing