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Particle Physics II – CP violation Lecture 2

Particle Physics II – CP violation Lecture 2. N. Tuning. Acknowledgements: Slides based on the course from Marcel Merk and Wouter Verkerke . Outline. 5 March Introduction: matter and anti-matter P, C and CP symmetries K-system CP violation Oscillations Cabibbo-GIM mechanism 12 March

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Particle Physics II – CP violation Lecture 2

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  1. Particle Physics II – CP violationLecture 2 N. Tuning Acknowledgements: Slides based on the course from Marcel Merk and Wouter Verkerke. Niels Tuning (1)

  2. Outline • 5 March • Introduction: matter and anti-matter • P, C and CP symmetries • K-system • CP violation • Oscillations • Cabibbo-GIM mechanism • 12 March • CP violation in the Lagrangian • CKM matrix • B-system • 19 March • B-factories • BJ/Psi Ks • Delta ms • (26 March: No lecture) • 2 April: • B-experiments: BaBar and LHCb • Measurements at LHCb Niels Tuning (2)

  3. Literature • Slides based on courses from Wouter Verkerke and Marcel Merk. • W.E. Burcham and M. Jobes, Nuclear and Particle Physics, chapters 11 and 14. • Z. Ligeti, hep-ph/0302031, Introduction to Heavy Meson Decays and CP Asymmetries • Y. Nir, hep-ph/0109090, CP Violation – A New Era • H. Quinn, hep-ph/0111177, B Physics and CP Violation Niels Tuning (3)

  4. Recap from last week Intrinsic spin +   P C + +  +   CP The Weak force and C,P parity violation • What about C+P  CP symmetry? • CP symmetry is parity conjugation (x,y,z  -x,-y,z) followed by charge conjugation (X  X) 100% P violation: All ν’s are lefthanded Allν’s are righthanded CP appears to be preservedin weakinteraction! Niels Tuning (4)

  5. Intermezzo: What operator is C ?Full solutions of Dirac equations: Try “ansatz”: 4 independent solutions for the Dirac spinors: Niels Tuning (5)

  6. Intermezzo: What operator is C ? Particle  Anti-particle: C=iγ2γ0 Dirac equation: In EM field: Possible choice for Cγ0 : (in Dirac representation) Cγ0= iγ2 Flip charge: e → -e Positron: Try out on a example spinor (e.g. electron with pos. helicity): Niels Tuning (6)

  7. Recap from last week The Cronin & Fitch experiment Essential idea: Look for K2 pp decays20 meters away from K0 production point Decay pions K2 ppdecays(CP Violation!) Incoming K2 beam K2 ppp decays Result: an excess of events at Q=0 degrees! Note scale: 99.99% of K ppp decaysare left of plot boundary Niels Tuning (7)

  8. Recap from last week Kaons: K0,K0, K1, K2, KS, KL, … • The kaons are produced in mass eigenstates: • | K0>: sd • |K0>: ds • The CP eigenstates are: • CP=+1: |K1> = 1/2 (|K0> - |K0>) • CP= -1: |K2> = 1/2 (|K0> + |K0>) • The kaons decay as short-lived or long-lived kaons: • |KS>: predominantly CP=+1 • |KL>: predominantly CP= -1 • η+-= (2.236 ± 0.007) x 10-3 • |ε| = (2.232 ± 0.007) x 10-3 Niels Tuning (8)

  9. Recap from last week Schematic picture of selected weak decays • K0 K0 transition • Note 1: Two W bosons required (DS=2 transition) • Note 2: many vertices, but still lowest order process… K2 Anti-K0 s d |K> W u u K0 K0 K0 W s d K1 Niels Tuning (9)

  10. Recap from last week The Cabibbo-GIM mechanism • How does it solve the K0 m+m- 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 s d s d -sinqc cosqc cosqc +sinqc c u nm nm m- m+ m- m+ Niels Tuning (10)

  11. Recap from last week From 2 to 3 generations • 2 generations: d’=0.97 d + 0.22 s (θc=13o) • 3 generations: d’=0.97 d + 0.22 s + 0.003 b • NB: probabilities have to add up to 1: 0.972+0.222+0.0032=1 •  “Unitarity” ! Niels Tuning (11)

  12. Recap from last week 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 Niels Tuning (12)

  13. Let’s investigate the Lagrangian • Hold on… Niels Tuning (13)

  14. The Standard Model Lagrangian • LKinetic :• Introduce the massless fermion fields • • Require local gauge invariance  gives rise to existence of gauge bosons • LHiggs :• Introduce Higgs potential with <f> ≠ 0 • • Spontaneous symmetry breaking The W+, W-,Z0 bosons acquire a mass • LYukawa :• Ad hoc interactions between Higgs field & fermions Niels Tuning (14)

  15. Interaction rep. Hypercharge Y (=avg el.charge in multiplet) SU(3)C SU(2)L Left- handed generation index Q = T3 + Y Fields: Notation Fermions: with y = QL, uR, dR, LL, lR, nR Quarks: Under SU2: Left handed doublets Right hander singlets Leptons: Scalar field: Note: Interaction representation: standard model interaction is independent of generation number Niels Tuning (15)

  16. Q = T3 + Y Fields: Notation Explicitly: • The left handed quark doublet : • Similarly for the quark singlets: • The left handed leptons: • And similarly the (charged) singlets: Niels Tuning (16)

  17. :The Kinetic Part : Fermions + gauge bosons + interactions Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant under SU(3)CxSU(2)LxU(1)Y transformations. Start with the Dirac Lagrangian: Replace: Gam :8 gluons Wbm: weak bosons: W1, W2, W3 Bm: hyperchargeboson Fields: Generators: La : Gell-Mann matrices: ½ la(3x3) SU(3)C Tb : Pauli Matrices: ½ tb (2x2) SU(2)L Y : Hypercharge: U(1)Y For the remainder we only consider Electroweak: SU(2)L x U(1)Y Niels Tuning (17)

  18. uLI W+m g dLI : The Kinetic Part For example, the term with QLiIbecomes: Writing out only the weak part for the quarks: W+ = (1/√2) (W1+ i W2) W- = (1/√ 2) (W1– i W2) L=JmWm Niels Tuning (18)

  19. Broken Symmetry V(f) V(f) Symmetry f f ~ 246 GeV Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value Substitute: Procedure: • . • The W+,W-,Z0 bosons acquire mass • The Higgs boson H appears : The Higgs Potential And rewrite the Lagrangian (tedious): (The other 3 Higgs fields are “eaten” by the W, Z bosons) Niels Tuning (19)

  20. doublets singlet : The Yukawa Part Since we have a Higgs field we can add (ad-hoc) interactions between f and the fermions in a gauge invariant way. The result is: i, j : indices for the 3 generations! With: (The CP conjugate of f To be manifestly invariant under SU(2) ) are arbitrary complex matrices which operate in family space (3x3)  Flavour physics! Niels Tuning (20)

  21. : The Yukawa Part Writing the first term explicitly: Niels Tuning (21)

  22. : The Yukawa Part • The hermiticity of the Lagrangian implies that there are terms in pairs of the form: • However a transformation under CP gives: CP is conserved inLYukawaonly ifYij = Yij* and leaves the coefficientsYij and Yij*unchanged Niels Tuning (22)

  23. : The Yukawa Part There are 3 Yukawa matrices (in the case of massless neutrino’s): • Each matrix is 3x3 complex: • 27 real parameters • 27 imaginary parameters (“phases”) • many of the parameters are equivalent, since the physics described by one set of couplings is the same as another • It can be shown (see ref. [Nir]) that the independent parameters are: • 12 real parameters • 1 imaginary phase • This single phase is the source of all CP violation in the Standard Model ……Revisit later Niels Tuning (23)

  24. S.S.B : The Fermion Masses Start with the Yukawa Lagrangian After which the following mass term emerges: with LMass is CP violating in a similar way as LYuk Niels Tuning (24)

  25. S.S.B dLI , uLI , lLIare the weak interaction eigenstates dL , uL , lLare the mass eigenstates (“physical particles”) : The Fermion Masses Writing in an explicit form: The matrices M can always be diagonalised by unitarymatricesVLfandVRfsuch that: Then the real fermion mass eigenstates are given by: Niels Tuning (25)

  26. S.S.B : The Fermion Masses In terms of the mass eigenstates: In flavour space one can choose: Weak basis: The gauge currents are diagonal in flavour space, but the flavour mass matrices are non-diagonal Mass basis: The fermion masses are diagonal, but some gauge currents (charged weak interactions) are not diagonal in flavour space In the weak basis: LYukawa = CP violating In the mass basis: LYukawa → LMass= CP conserving  What happened to the charged current interactions (in LKinetic) ? Niels Tuning (26)

  27. : The Charged Current The charged current interaction for quarks in the interaction basis is: The charged current interaction for quarks in the mass basis is: The unitary matrix: With: is the Cabibbo Kobayashi Maskawa mixing matrix: Lepton sector: similarly However, for massless neutrino’s: VLn= arbitrary. Choose it such that VMNS = 1 There is no mixing in the lepton sector Niels Tuning (27)

  28. Charged Currents The charged current term reads: Under the CP operator this gives: (Together with (x,t) -> (-x,t)) A comparison shows that CP is conserved only ifVij = Vij* In general the charged current term is CP violating Niels Tuning (28)

  29. W+ W- b gVub gV*ub u Why complex phases matter • CP conjugation of a W boson vertex involves complex conjugation of coupling constant • With 2 generations Vij is always real and VijVij* • With 3 generations Vij can be complex CP violation built into weak decay mechanism! Above process violates CP if Vub Vub* Niels Tuning (29)

  30. The Standard Model Lagrangian (recap) • LKinetic : •Introduce the massless fermion fields • •Require local gauge invariance  gives rise to existence of gauge bosons  CP Conserving • LHiggs : •Introduce Higgs potential with <f> ≠ 0 • •Spontaneous symmetry breaking The W+, W-,Z0 bosons acquire a mass  CP Conserving • LYukawa : •Ad hoc interactions between Higgs field & fermions  CP violating with a single phase • LYukawa → Lmass : • fermion weak eigenstates: • - mass matrix is (3x3) non-diagonal • • fermion mass eigenstates: • - mass matrix is (3x3) diagonal  CP-violating  CP-conserving! • LKinetic in mass eigenstates: CKM – matrix  CP violating with a single phase Niels Tuning (30)

  31. 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 (0.972+0.222+0.0032=1) • How many free parameters? • How many real/complex? • How do we normally visualize these parameters? Niels Tuning (31)

  32. 2 generations: No CP violation in SM! This is the reason Kobayashi and Maskawa first suggested a 3rd family of fermions! Quark field re-phasing Under a quark phase transformation: and a simultaneous rephasing of the CKM matrix: or the charged current is left invariant. Degrees of freedom in VCKM in 3 N generations Number of real parameters: 9+ N2 Number of imaginary parameters: 9 + N2 Number of constraints (VV† = 1): -9- N2 Number of relative quark phases: -5- (2N-1) ----------------------- Total degrees of freedom: 4(N-1)2 Number of Euler angles: 3N (N-1) / 2 Number of CP phases: 1 (N-1) (N-2) / 2 Niels Tuning (32)

  33. 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 • 4 parameters • 3 real • 1 phase Magnitude of elements shown only, no information of phase Niels Tuning (33)

  34. d s b u c t Approximately diagonal form • Values are strongly ranked: • Transition within generation favored • Transition from 1st to 2nd generation suppressed by cos(qc) • Transition from 2nd to 3rd generation suppressed bu cos2(qc) • Transition from 1st to 3rd generation suppressed by cos3(qc) CKM magnitudes Why the ranking?We don’t know (yet)! If you figure this out,you will win the nobelprize l l3 l l2 l3 l2 l=cos(qc)=0.23 Niels Tuning (34)

  35. Exploit apparent ranking for a convenient parameterization • Given current experimental precision on CKM element values, we usually drop l4 and l5 terms as well • Effect of order 0.2%... • Deviation of ranking of 1st and 2nd generation (l vs l2) parameterized in A parameter • Deviation of ranking between 1st and 3rd generation, parameterized through |r-ih| • Complex phase parameterized in arg(r-ih) Niels Tuning (35)

  36. ~1995 What do we know about A, λ, ρ and η? • Fit all known Vij values to Wolfenstein parameterization and extract A, λ, ρ and η • Results for A and l most precise (but don’t tell us much about CPV) • A = 0.83, l = 0.227 • Results for r,h are usually shown in complex plane of r-ih for easier interpretation Niels Tuning (36)

  37. Deriving the triangle interpretation • Starting point: the 9 unitarity constraints on the CKM matrix • Pick (arbitrarily) orthogonality condition with (i,j)=(3,1) Niels Tuning (37)

  38. Visualizing the unitarity constraint • Sum of three complex vectors is zero  Form triangle when put head to tail (Wolfenstein params to order l4) Niels Tuning (38)

  39. Visualizing the unitarity constraint • Phase of ‘base’ is zero  Aligns with ‘real’ axis, Niels Tuning (39)

  40. Visualizing the unitarity constraint • Divide all sides by length of base • Constructed a triangle with apex (r,h) (r,h) (0,0) (1,0) Niels Tuning (40)

  41. Visualizing arg(Vub) and arg(Vtd) in the (r,h) plane • We can now put this triangle in the (r,h) plane Niels Tuning (41)

  42. Outline • 5 March • Introduction: matter and anti-matter • P, C and CP symmetries • K-system • CP violation • Oscillations • Cabibbo-GIM mechanism • 12 March • CP violation in the Lagrangian • CKM matrix • B-system • 19 March • B-factories • BJ/Psi Ks • Delta ms • (26 March: No lecture) • 2 April: • B-experiments: BaBar and LHCb • Measurements at LHCb Niels Tuning (42)

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