July 1-2, The Origin of CP Violation in the Standard Model Topical Lectures July 1-2, 2004 Marcel Merk

July 1-2, Contents Introduction: symmetry and non-observables CPT Invariance CP Violation in the Standard Model Lagrangian Re-phasing independent CP Violation quantities The Fermion masses The matter anti-matter asymmetry Theory Oriented!

July 1-2, Literature C.Jarlskog, “Introduction to CP Violation”, Advanced Series on Directions in High Energy Physics – Vol 3: “CP Violation’, 1998, p3. Y.Nir, “CP Violation In and Beyond the Standard Model”, Lectures given at the XXVII SLAC Summer Institute, hep-ph/ Branco, Lavoura, Silva: “CP Violation”, International series of monographs on physics, Oxford univ. press, Bigi and Sanda: “CP Violation”, Cambridge monographs on particle physics, nuclear physics and cosmology, Cambridge univ. press, T.D. Lee, “Particle Physics and Introduction to Field Theory”, Contemporary Concepts in Physics Volume 1, Revised and Updated First Edition, Harwood Academic Publishers, C. Quigg, “Gauge Theories of the Strong, Weak and Electromagnetic Interactions”, Frontiers in Physics, Benjamin-Cummings, H. Fritsch and Z. Xing, “Mass and Flavour Mixing Schemes of Quarks and Leptons”, hep-ph/ Mark Trodden, “Electroweak Baryogenesis”, hep-ph/ References:

July 1-2, Introduction: Symmetry and non-Observables T.D.Lee: “The root to all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities; the non-observables” There are four main types of symmetry: Permutation symmetry: Bose-Einstein and Fermi-Dirac Statistics Continuous space-time symmetries: translation, rotation, acceleration,… Discrete symmetries: space inversion, time inversion, charge inversion Unitary symmetries: gauge invariances: U 1 (charge), SU 2 (isospin), SU 3 (color),..  If a quantity is fundamentally non-observable it is related to an exact symmetry  If a quantity could in principle be observed by an improved measurement; the symmetry is said to be broken Noether Theorem:symmetryconservation law

July 1-2, Symmetry and non-observables Simple Example: Potential energy V between two particles: Absolute position is a non-observable: The interaction is independent on the choice of 0. Symmetry: V is invariant under arbitrary space translations: Consequently: Total momentum is conserved: 00’

July 1-2, Symmetry and non-observables Non-observablesSymmetry TransformationsConservation Laws or Selection Rules Difference between identical particles PermutationB.-E. or F.D. statistics Absolute spatial positionSpace translationmomentum Absolute timeTime translationenergy Absolute spatial directionRotationangular momentum Absolute velocityLorentz transformationgenerators of the Lorentz group Absolute right (or left)parity Absolute sign of electric chargecharge conjugation Relative phase between states of different charge Q charge Relative phase between states of different baryon number B baryon number Relative phase between states of different lepton number L lepton number Difference between different co- herent mixture of p and n states isospin

July 1-2, Parity Violation Before 1956 physicists were convinced that the laws of nature were left-right symmetric. Strange? A “gedanken” experiment: Consider two perfectly mirror symmetric cars: “L” and “R” are fully symmetric, Each nut, bolt, molecule etc. However the engine is a black box Person “L” gets in, starts, ….. 60 km/h Person “R” gets in, starts, ….. What happens? What happens in case the ignition mechanism uses, say, Co 60  decay? “L” “R” Gas pedal driver Gas pedal driver

July 1-2, CPT Invariance Local Field theories always respect: Lorentz Invariance Symmetry under CPT operation (an electron = a positron travelling back in time) => Consequence: mass of particle = mass of anti-particle: Question 1: The mass difference between K L and K S :  m = 3.5 x eV => CPT violation? Question 2: How come the lifetime of K S = ns while the lifetime of the K L = 51.7 ns? Question 3: BaBar measures decay rate B-> J/  K S and Bbar-> J/  K S. Clearly not the same: how can it be? => Similarly the total decay-rate of a particle is equal to that of the anti-particle Answer 3: Partial decay rate ≠ total decay rate! However, the sum over all partial rates (>200 or so) is the same for B and Bbar. (Amazing! – at least to me) Answer 1 + 2: A K L ≠ an anti- K S particle! (Lüders, Pauli, Schwinger) (anti-unitarity)

July 1-2, L SM contains: L Kinetic : fermion fields L Higgs : the Higgs potential L Yukawa : the Higgs – Fermion interactions Plan: Look at symmetry aspects of the Lagrangian How is CP violation implemented? → Several “miracles” happen in symmetry breaking CP in the Standard Model Lagrangian (The origin of the CKM-matrix) Standard Model gauge symmetry: Note Immediately: The weak part is explicitly parity violating Outline: Lorentz structure of the Lagrangian Introduce the fermion fields in the SM L Kinetic : local gauge invariance : fermions ↔ bosons L Higgs : spontaneous symmetry breaking L Yukawa : the origin of fermion masses V CKM : CP violation

July 1-2, Lagrangian Density Local field theories work with Lagrangian densities: The fundamental quantity, when discussing symmetries is the Action: If the action is (is not) invariant under a symmetry operation then the symmetry in question is a good (broken) one => Unitarity of the interaction requires the Lagrangian to be Hermitian with the fields taken at

July 1-2, Structure of a Lagrangian Example: Consider a spin-1/2 (Dirac) particle (“nucleon”) interacting with a spin-0 (Scalar) object (“meson”) Meson potential Nucleon field Nucleon – meson interaction Lorentz structure: a Lagrangian in field theory can be built using combinations of: S: Scalar fields : 1 P: Pseudoscalar fields :  5 V: Vector fields :   A: Axial vector fields :    5 T: Tensor fields :    Dirac field  Exercise: What are the symmetries of this theory under C, P, CP ? Can a and b be any complex numbers? Note: the interaction term contains scalar and pseudoscalar parts Scalar field  Violates P, conserves C, violates CP a and b must be real from Hermeticity

July 1-2, Transformation Properties Transformation properties of Dirac spinor bilinears: (Ignoring arbitrary phases) c→c *

July 1-2, The Standard Model Lagrangian L Kinetic : Introduce the massless fermion fields Require local gauge invariance => gives rise to existence of gauge bosons L Higgs : Introduce Higgs potential with ≠ 0 Spontaneous symmetry breaking L Yukawa : Ad hoc interactions between Higgs field & fermions L Yukawa → L mass : fermion weak eigenstates: -- mass matrix is (3x3) non-diagonal fermion mass eigenstates: -- mass matrix is (3x3) diagonal L Kinetic in mass eigenstates: CKM – matrix The W +, W -,Z 0 bosons acquire a mass => CP Conserving => CP violating with a single phase => CP-violating => CP-conserving! => CP violating with a single phase

July 1-2, Fields: Notation SU(3) C SU(2) L Y Left- handed generation index Interaction rep. Quarks: Leptons: Scalar field: Q = T 3 + Y Under SU2: Left handed doublets Right hander singlets Note: Interaction representation: standard model interaction is independent of generation number Fermions: with  = Q L, u R, d R, L L, l R, R

July 1-2, Fields: Notation Explicitly: Similarly for the quark singlets: And similarly the (charged) singlets: The left handed leptons: The left handed quark doublet : Q = T 3 + Y

July 1-2, Intermezzo: Local Gauge Invariance in a single transparancy Basic principle: The Lagrangian must be invariant under local gauge transformations Example: massless Dirac Spinors in QED: “global” U(1) gauge transformation: “local” U(1) gauge transformation: Is the Lagrangian invariant? Then: Not invariant! => Introduce the covariant derivative: and demand that   transforms as: Then it turns out that: Introduce charged fermion field (electron) Demand invariance under local gauge transformations (U(1)) The price to pay is that a gauge field A  must be introduced at the same time (the photon) is invariant! Conclusion:

July 1-2, Fermions + gauge bosons + interactions Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant Start with the Dirac Lagrangian: Replace: Fields: Generators: a  : G a  : 8 gluons W b   : weak bosons: W 1, W 2, W 3 B  : B  : hypercharge boson L a : Gell-Mann matrices: ½ a (3x3) SU(3) C T b : Pauli Matrices: ½  b (2x2) SU(2) L Y : Hypercharge: U(1) Y :The Kinetic Part For the remainder we only consider Electroweak: SU(2) L x U(1) Y

July 1-2, : The Kinetic Part Exercise: Show that this Lagrangian formally violates both P and C Show that this Lagrangian conserves CP For example the term with Q Li I becomes: and similarly for all other terms ( u Ri I,d Ri I,L Li I,l Ri I ). Writing out only the weak part for the quarks: uLIuLI dLIdLI g W+W+ L Kin = CP conserving W + = (1/√2) (W 1 + i W 2 ) W - = (1/√ 2) (W 1 – i W 2 ) L=JWL=JW

July 1-2, : The Higgs Potential →Note L Higgs = CP conserving V   V(  )  SymmetryBroken Symmetry ~ 246 GeV Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value Procedure:Substitute: And rewrite the Lagrangian (tedious): 1. 2.The W +,W -,Z 0 bosons acquire mass 3.The Higgs boson H appears “The realization of the vacuum breaks the symmetry” (The other 3 Higgs fields are “eaten” by the W, Z bosons)

July 1-2, : The Yukawa Part Since we have a Higgs field we can add (ad-hoc) interactions between  and the fermions in a gauge invariant way. L must be Her- mitian (unitary) The result is: are arbitrary complex matrices which operate in family space (3x3) => Flavour physics! doublets singlet With: To be manifestly invariant under SU(2)

July 1-2, : The Yukawa Part Writing the first term explicitly: Question: In what aspect is this Lagrangian similar to the example of the nucleon-meson potential?

July 1-2, : The Yukawa Part Exercise (intuitive proof) Show that: The hermiticity of the Lagrangian implies that there are terms in pairs of the form: However a transformation under CP gives: and leaves the coefficients Y ij and Y ij * unchanged CP is conserved in L Yukawa only if Y ij = Y ij * In general L Yukawa is CP violating Formally, CP is violated if:

July 1-2, : 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

July 1-2, : The Fermion Masses S.S.B Start with the Yukawa Lagrangian After which the following mass term emerges: with L Mass is CP violating in a similar way as L Yuk

July 1-2, : The Fermion Masses S.S.B Writing in an explicit form: The matrices M can always be diagonalised by unitary matrices V L f and V R f such that: Then the real fermion mass eigenstates are given by: d L I, u L I, l L I are the weak interaction eigenstates d L, u L, l L are the mass eigenstates (“physical particles”)

July 1-2, : The Fermion Masses S.S.B In terms of the mass eigenstates: = CP Conserving? 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: L Yukawa = CP violating In the mass basis: L Yukawa → L Mass = CP conserving =>What happened to the charged current interactions (in L Kinetic ) ?

July 1-2, : 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: is the Cabibbo Kobayashi Maskawa mixing matrix: With: Lepton sector: similarly However, for massless neutrino’s: V L = arbitrary. Choose it such that V MNS = 1 => There is no mixing in the lepton sector

July 1-2, Flavour Changing Neutral Currents To illustrate the SM neutral current take the W 3  and B  term of the Kinetic Lagrangian: In terms of physical fields no non-diagonal contributions occur for the neutral Currents. => GIM mechanism And consider the Z-boson field: Take further Q Li I =d Li I Use: Standard Model forbids flavour changing neutral currents. and

July 1-2, Charged Currents A comparison shows that CP is conserved only if V ij = V ij * (Together with (x,t) -> (-x,t)) The charged current term reads: Under the CP operator this gives: In general the charged current term is CP violating

July 1-2, Charged Currents A comparison shows that CP is conserved only if V ij = V ij * (Together with (x,t) -> (-x,t)) The charged current term reads: Under the CP operator this gives: In general the charged current term is CP violating

July 1-2, Where were we?

July 1-2, The Standard Model Lagrangian (recap) L Kinetic : Introduce the massless fermion fields Require local gauge invariance => gives rise to existence of gauge bosons L Higgs : Introduce Higgs potential with ≠ 0 Spontaneous symmetry breaking L Yukawa : Ad hoc interactions between Higgs field & fermions L Yukawa → L mass : fermion weak eigenstates: -- mass matrix is (3x3) non-diagonal fermion mass eigenstates: -- mass matrix is (3x3) diagonal L Kinetic in mass eigenstates: CKM – matrix The W +, W -,Z 0 bosons acquire a mass => CP Conserving => CP violating with a single phase => CP-violating => CP-conserving! => CP violating with a single phase

July 1-2, 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 third family of fermions! 2 generations:

July 1-2, The LEP CERN Maybe the most important result of LEP: “There are 3 generations of neutrino’s” L3L3 Aleph Opal Delphi Geneva Airport “Cointrin” MZMZ Light, left-handed, “active”

July 1-2, The lepton sector (Intermezzo) N. Cabibbo: Phys. Rev.Lett. 10, 531 (1963) –2 family flavour mixing in quark sector (GIM mechanism) M.Kobayashi and T.Maskawa, Prog. Theor. Phys 49, 652 (1973) –3 family flavour mixing in quark sector Z.Maki, M.Nakagawa and S.Sakata, Prog. Theor. Phys. 28, 870 (1962) –2 family flavour mixing in neutrino sector to explain neutrino oscillations! In case neutrino masses are of the Dirac type, the situation in the lepton sector is very similar as in the quark sector: V MNS ~ V CKM. –The is one CP violating phase in the lepton MNS matrix In case neutrino masses are of the Majorana type (a neutrino is its own anti- particle → no freedom to redefine neutrino phases) –There are 3 CP violating phases in the lepton MNS matrix However, the two extra phases are unobservable in neutrino oscillations –There is even a CP violating phase in case N dim = 2

July 1-2, Lepton mixing and neutrino oscillations In the CKM we write by convention the mixing for the down type quarks; in the lepton sector we write it for the (up-type) neutrinos. Is it relevant? –If yes: why? –If not, why don’t we measure charged lepton oscillations rather then neutrino oscillations? lLIlLI νLIνLI W+W+ However, observation of neutrino oscillations is possible due to small neutrino mass differences. Question: e   W

July 1-2, Rephasing Invariants Simplest: U  i = |V  i | 2 is independent of quark re-phasing Next simplest: Quartets: Q  i  j = V  i V  j V  j * V  i * with  ≠  and i≠j –“Each quark phase appears with and without *” V † V=1 : Unitarity triangle: V ud V cd * + V us V cs * + V ub V cb * = 0 –Multiply the equation by V us * V cs and take the imaginary part: –Im (V us * V cs V ud V cd * ) = - Im (V us * V cs V ub V cb * ) –J = Im Q udcs = - Im Q ubcs –The imaginary part of each Quartet combination is the same (up to a sign) –In fact it is equal to 2x the surface of the unitarity triangle Im[V  i V  j V  j * V  i *] = J ∑    ijk where J is the universal Jarlskog invariant Amount of CP Violation is proportional to J The standard representation of the CKM matrix is: However, many representations are possible. What are the invariants under re-phasing?

July 1-2, unitarity: The Unitarity Triangle    V cd V cb * V td V tb * V ud V ub * Under re-phasing:the unitary angles are invariant (In fact, rephasing implies a rotation of the whole triangle) Area = ½ | Im Q udcb | = ½ | J | The “ db ” triangle: V CKM † V CKM = 1

July 1-2, Wolfenstein Parametrization Wolfenstein realised that the non-diagonal CKM elements are relatively small compared to the diagonal elements, and parametrized as follows: Normalised CKM triangle:    (0,0)(1,0) 

July 1-2, CP Violation and quark masses Note that the massless Lagrangian has a global symmetry for unitary transformations in flavour space. Let’s now assume two quarks with the same charge are degenerate in mass, eg.: m s = m b Redefine: s’ = V us s + V ub b Now the u quark only couples to s’ and not to b’ : i.e. V 13 ’ = 0 Using unitarity we can show that the CKM matrix can now be written as: CP conserving Necessary criteria for CP violation:

July 1-2, The Amount of CP Violation However, also required is: All requirements for CP violation can be summarized by: (The maximal value J might have = 1/(6√3) ~ 0.1) Using Standard Parametrization of CKM: Is CP violation maximal? => One has to understand the origin of mass! (eg.: J=Im(V us V cb V ub * V cs * ) )

July 1-2, Mass Patterns Mass spectra (  = M z, MS-bar scheme) m u ~ MeV, m c ~ 0.5 – 0.6 GeV, m t ~ 180 GeV m d ~ MeV, m s ~ 35 – 100 MeV, m b ~ 2.9 GeV Why are neutrino’s so light? Related to the fact that they are the only neutral fermions? See-saw mechanism? m e = 0.51 MeV, m  = 105 MeV, m  = 1777 MeV Do you want to be famous? Do you want to be a king? Do you want more then the nobel prize? - Then solve the mass Problem – R.P. Feynman Observe:

July 1-2, Matter - antimatter asymmetry In the visible universe matter dominates over anti-matter: There are no antimatter particles present in cosmic rays There are no intense  -ray sources in the universe due to matter anti-matter collisions Hubble deep field - optical

July 1-2, Big Bang Cosmology Equal amounts of matter & antimatter q+q ⇄  +  Matter Dominates ! + CMB

July 1-2, The matter anti-matter asymmetry Almost all matter annihilated with antimatter, producing photons… WMAP satellite K2.7252K Cosmic Microwave Background Angular Power Spectrum

July 1-2, A matter dominated universe can evolve in case three conditions occur simultaneous: 1)Baryon number violation: L(  B)≠0 2)C and CP Violation:  (N→f) ≠  (N→f) 3)Thermal non-equilibrium: otherwise: CPT invariance => CP invariance The Sakharov conditions Sakharov (1964) Convert 1 in 10 9 anti-quarks into a quark in an early stage of universe: Anti-

July 1-2, Baryogenesis at the GUT Scale d u u X e+e+ u u - proton 00 GUT theories predict proton decay mediated by heavy X gauge bosons: X boson has baryon number violating (1) couplings: X →q q, X→q l Proton lifetime:    s Decay process Decay fraction BB X → q qr2/3 X → q l1-r1/3 X → q qr-2/3 X → q l1-r-1/3 Efficiency  of Baryon asymmetry build-up: A simple Baryogenesis model: CP Violation (2) : r ≠ r Assuming the back reaction does not occur (3): Initial X number density Initial light particle number density Conceptually simple

July 1-2, Baryogenesis at Electroweak Scale SM Electroweak Interactions: 1) Baryon number violation in weak anomaly: Conserves “B-L” but violates “B+L” 2) CP Violation in the CKM 3) Non-equilibrium: electroweak phase transition Conceptually difficult Electroweak phase transition wipes out GUT Baryon asymmetry! Problems: 1. Higgs mass is too heavy. In order to have a first order phase transition: Requirement: m H ~ 100 GeV/c 2 2. CP Violation in CKM is not enough: Leptogenesis: Uses the large right handed majorana neutrino masses in the see-saw mechanism to generate a lepton asymmetry at high energies (using the MNS equivalent of CKM). Uses the electroweak sphaleron (“B-L” conserving) processes to communicate this to a baryon asymmetry, which survives further evolution of the universe. Can it generate a sufficiently large asymmetry?

July 1-2, Conclusion Biertje? Key questions in B physics: Is the SM the only source of CP Violations? Does the SM fully explain flavour physics?