1. Niels Tuning (1) CP violation Lecture 2 N. Tuning

3. Recap: Interesting because: 1)Standard Model: in the heart of quark interactions 2)Cosmology: related to matter – anti-matter asymetry 3)Beyond Standard Model: measurements are sensitive to new particles Niels Tuning (3) CP-violation (or flavour physics) is about charged current interactions b s s b Matter Dominates !

4. Diagonalize Yukawa matrix Y ij –Mass terms –Quarks rotate –Off diagonal terms in charged current couplings Niels Tuning (4) Recap uIuI dIdI W u d,s,b W

5. Niels Tuning (5) Ok…. We’ve got the CKM matrix, now what? It’s unitary –“probabilities add up to 1”: –d’=0.97 d + 0.22 s + 0.003 b (0.97 2 +0.22 2 +0.003 2 =1) How many free parameters? –How many real/complex? How do we normally visualize these parameters?

6. Niels Tuning (6) Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or In other words:

7. Niels Tuning (7) Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or the charged currentis left invariant. Degrees of freedom in V CKM in 3 N generations Number of real parameters: 9 + N 2 Number of imaginary parameters: 9 + N 2 Number of constraints ( VV † = 1): -9 - N 2 Number of relative quark phases: -5 - (2N-1) ----------------------- Total degrees of freedom: 4 (N-1) 2 Number of Euler angles: 3 N (N-1) / 2 Number of CP phases: 1 (N-1) (N-2) / 2 No CP violation in SM! This is the reason Kobayashi and Maskawa first suggested a 3 rd family of fermions! 2 generations:

8. Intermezzo: Kobayashi & Maskawa Niels Tuning (8)

9. Timeline: Niels Tuning (9) Timeline: –Sep 1972: Kobayashi & Maskawa predict 3 generations –Nov 1974: Richter, Ting discover J/ ψ: fill 2 nd generation –July 1977: Ledermann discovers Υ : discovery of 3 rd generation

10. Niels Tuning (10) Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or the charged currentis left invariant. Degrees of freedom in V CKM in 3 N generations Number of real parameters: 9 + N 2 Number of imaginary parameters: 9 + N 2 Number of constraints ( VV † = 1): -9 - N 2 Number of relative quark phases: -5 - (2N-1) ----------------------- Total degrees of freedom: 4 (N-1) 2 Number of Euler angles: 3 N (N-1) / 2 Number of CP phases: 1 (N-1) (N-2) / 2 No CP violation in SM! This is the reason Kobayashi and Maskawa first suggested a 3 rd family of fermions! 2 generations:

11. Niels Tuning (11) Cabibbos theory successfully correlated many decay rates Cabibbos theory successfully correlated many decay rates by counting the number of cos c and sin c terms in their decay diagram

12. Niels Tuning (12) Cabibbos theory successfully correlated many decay rates There was however one major exception which Cabibbo could not describe: K 0   +  - –Observed rate much lower than expected from Cabibbos rate correlations (expected rate  g 8 sin 2  c cos 2  c ) d ++ --  u cos c sin c W W ss

13. Niels Tuning (13) The Cabibbo-GIM mechanism Solution to K 0 decay problem in 1970 by Glashow, Iliopoulos and Maiani  postulate existence of 4 th quark –Two ‘up-type’ quarks decay into rotated ‘down-type’ states –Appealing symmetry between generations u d’=cos( c )d+sin( c )s W+W+ c s’=-sin( c )d+cos(c)s W+W+

14. Niels Tuning (14) The Cabibbo-GIM mechanism How does it solve the K 0   +  - problem? –Second decay amplitude added that is almost identical to original one, but has relative minus sign  Almost fully destructive interference –Cancellation not perfect because u, c mass different d ++ --  u cos c +sin c d ++ --  c -sin c cos c ss ss

15. Niels Tuning (15) From 2 to 3 generations 2 generations: d’=0.97 d + 0.22 s (θ c =13 o ) 3 generations: d’=0.97 d + 0.22 s + 0.003 b NB: probabilities have to add up to 1: 0.97 2 +0.22 2 +0.003 2 =1 –  “Unitarity” !

16. From 2 to 3 generations 2 generations: d’=0.97 d + 0.22 s (θ c =13 o ) 3 generations: d’=0.97 d + 0.22 s + 0.003 b Parameterization used by Particle Data Group (3 Euler angles, 1 phase):

17. Niels Tuning (17) Possible forms of 3 generation mixing matrix ‘General’ 4-parameter form (Particle Data Group) with three rotations  12, 13, 23 and one complex phase  13 –c 12 = cos( 12 ), s 12 = sin( 12 ) etc… Another form (Kobayashi & Maskawa’s original) –Different but equivalent Physics is independent of choice of parameterization! –But for any choice there will be complex-valued elements

18. Niels Tuning (18) Possible forms of 3 generation mixing matrix  Different parametrizations! It’s about phase differences! Re-phasing V: PDG KM 3 parameters: θ, τ, σ 1 phase : φ

19. Niels Tuning (19) How do you measure those numbers? Magnitudes are typically determined from ratio of decay rates Example 1 – Measurement of V ud –Compare decay rates of neutron decay and muon decay –Ratio proportional to V ud 2 –|V ud | = 0.97425 ± 0.00022 –V ud of order 1

20. How do you measure those numbers? Example 2 – Measurement of V us –Compare decay rates of semileptonic K- decay and muon decay –Ratio proportional to V us 2 –|V us | = 0.2252 ± 0.0009 –V us  sin( c )

21. How do you measure those numbers? Example 3 – Measurement of V cb –Compare decay rates of B 0  D *- l + and muon decay –Ratio proportional to V cb 2 –|V cb | = 0.0406 ± 0.0013 –V cb is of order sin( c ) 2 [= 0.0484]

22. How do you measure those numbers? Example 4 – Measurement of V ub –Compare decay rates of B 0  D *- l + and B 0   - l + –Ratio proportional to (V ub /V cb) 2 –|V ub /V cb | = 0.090 ± 0.025 –V ub is of order sin( c ) 3 [= 0.01]

23. How do you measure those numbers? Example 5 – Measurement of V cd –Measure charm in DIS with neutrinos –Rate proportional to V cd 2 –|V cd | = 0.230 ± 0.011 –V cb is of order sin( c ) [= 0.23]

24. How do you measure those numbers? Example 6 – Measurement of V tb –Very recent measurement: March ’09! –Single top production at Tevatron –CDF+D0: |V tb | = 0.88 ± 0.07

25. How do you measure those numbers? Example 7 – Measurement of V td, V ts –Cannot be measured from top-decay… –Indirect from loop diagram –V ts : recent measurement: March ’06 –|V td | = 0.0084 ± 0.0006 –|V ts | = 0.0387 ± 0.0021 V ts Ratio of frequencies for B 0 and B s  = 1.210 +0.047 from lattice QCD -0.035 V ts ~ 2 V td ~ 3   Δm s ~ (1/ λ 2 ) Δm d ~ 25 Δm d

26. Niels Tuning (26) What do we know about the CKM matrix? Magnitudes of elements have been measured over time –Result of a large number of measurements and calculations Magnitude of elements shown only, no information of phase

27. Niels Tuning (27) What do we know about the CKM matrix? Magnitudes of elements have been measured over time –Result of a large number of measurements and calculations Magnitude of elements shown only, no information of phase

28. Niels Tuning (28) Approximately diagonal form Values are strongly ranked: –Transition within generation favored –Transition from 1 st to 2 nd generation suppressed by cos( c ) –Transition from 2 nd to 3 rd generation suppressed bu cos 2 ( c ) –Transition from 1 st to 3 rd generation suppressed by cos 3 ( c ) u d t c bs CKM magnitudes 3 2 2 3 =sin( c )=0.23 Why the ranking? We don’t know (yet)! If you figure this out, you will win the nobel prize

29. Intermezzo: How about the leptons? We now know that neutrinos also have flavour oscillations thus there is the equivalent of a CKM matrix for them: –Pontecorvo-Maki-Nakagawa-Sakata matrix a completely different hierarchy! Niels Tuning (29) vs

30. Wolfenstein parameterization 3 real parameters: A, λ, ρ 1 imaginary parameter : η

31. Wolfenstein parameterization 3 real parameters: A, λ, ρ 1 imaginary parameter : η

32. Niels Tuning (32) Exploit apparent ranking for a convenient parameterization Given current experimental precision on CKM element values, we usually drop 4 and 5 terms as well –Effect of order 0.2%... Deviation of ranking of 1 st and 2 nd generation ( vs 2 ) parameterized in A parameter Deviation of ranking between 1 st and 3 rd generation, parameterized through |-i| Complex phase parameterized in arg(-i)

33. Niels Tuning (33) ~1995 What do we know about A, λ, ρ and η ? Fit all known V ij values to Wolfenstein parameterization and extract A, λ, ρ and η Results for A and most precise (but don’t tell us much about CPV) –A = 0.83, = 0.227 Results for  are usually shown in complex plane of -i for easier interpretation

34. Niels Tuning (34) Deriving the triangle interpretation Starting point: the 9 unitarity constraints on the CKM matrix Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

35. Niels Tuning (35) Deriving the triangle interpretation Starting point: the 9 unitarity constraints on the CKM matrix –3 orthogonality relations Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

36. Niels Tuning (36) Deriving the triangle interpretation Starting point: the 9 unitarity constraints on the CKM matrix Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

37. Niels Tuning (37) Visualizing the unitarity constraint Sum of three complex vectors is zero  Form triangle when put head to tail (Wolfenstein params to order 4 )

38. Niels Tuning (38) Visualizing the unitarity constraint Phase of ‘base’ is zero  Aligns with ‘real’ axis,

39. Niels Tuning (39) Visualizing the unitarity constraint Divide all sides by length of base Constructed a triangle with apex (,) (0,0)(1,0) (,)(,)

40. Niels Tuning (40) Visualizing arg(V ub ) and arg(V td ) in the () plane We can now put this triangle in the () plane

41. “The” Unitarity triangle We can visualize the CKM-constraints in () plane

42. β We can correlate the angles β and γ to CKM elements:

43. Niels Tuning (43) Deriving the triangle interpretation Another 3 orthogonality relations Pick (arbitrarily) orthogonality condition with (i,j)=(3,1)

44. The “other” Unitarity triangle NB: angle β s introduced. But… not phase invariant definition!? Two of the six unitarity triangles have equal sides in O(λ) Niels Tuning (44)

45. The “B s -triangle”: β s Niels Tuning (45) Replace d by s:

46. The phases in the Wolfenstein parameterization Niels Tuning (46)

47. The CKM matrix Niels Tuning (47) Couplings of the charged current: Wolfenstein parametrization: Magnitude:Complex phases: b WW u gV ub

48. Niels Tuning (48) Back to finding new measurements Next order of business: Devise an experiment that measures arg(V td )and arg(V ub ). –What will such a measurement look like in the () plane?   CKM phases Fictitious measurement of  consistent with CKM model

49. Niels Tuning (49) Consistency with other measurements in (,) plane Precise measurement of sin(2β) agrees perfectly with other measurements and CKM model assumptions The CKM model of CP violation experimentally confirmed with high precision!

50. What’s going on?? ??? Edward Witten, 17 Feb 2009…17 Feb 2009 See “From F-Theory GUT’s to the LHC” by Heckman and Vafa (arXiv:0809.3452)