Part II CP Violation in the SM Chris Parkes

Outline THEORETICAL CONCEPTS Introductory concepts Matter and antimatter Symmetries and conservation laws Discrete symmetries P, C and T 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

discovery of CP violation Kaons and discovery of CP violation

 +  + +  +   C P CP What about the product CP? 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?  +  Intrinsic spin P C + +  Initially CP appears to be preserved in weak interactions …! +   CP

Requires higher energy Introducing kaons Kaon mesons: in two isospin doublets Part of pseudo-scalar JP=0- mesons octet with p, h K+ = us Ko = ds I3=+1/2 I3=-1/2 Ko = ds K- = us S=+1 S=-1 Kaon production: (pion beam hitting a target) Ko : - + p  o + Ko But from baryon number conservation: Ko : + + p  K+ + Ko + p Or Ko : - + p  o + Ko + n +n Requires higher energy S 0 0 -1 +1 S 0 0 +1 -1 0 Much higher S 0 0 +1 -1 0 0

Neutral kaons (1/2) What precisely is a K0 meson? Now we know the quark contents: K0 =sd, K0 =sd First: what is the effect of C and P on the K0 and K0 particles? (because l=0 q qbar pair) effect of CP : Bottom line: the flavour eigenstates K0 and K0 are not CP eigenstates (flavour eigenstates – definite quark flavour states)

Neutral kaons (2/2) Nevertheless it is possible to construct CP eigenstates as linear combinations Can always be done in quantum mechanics, to construct CP eigenstates |K1> = 1/2(|K0> + |K0>) |K2> = 1/2(|K0> - |K0>) Then: CP |K1> = +1 |K1> CP |K2> = -1 |K2> Does it make sense to look at these linear combinations? i.e. do these represent real particles? Predictions were: The K1 must decay to 2 pions assuming CP conservation of the weak interactions This 2 pion neutral kaon decay was the decay observed and therefore known The same arguments predict that K2 must decay to 3 pions History tells us it made sense! The K2 = KL (“K-long”) was discovered in 1956 after being predicted (actually not same - difference between K2 and KL due to CP violation to be discussed later)

Looking closer at KL decays How do you obtain a pure ‘beam’ of K2 particles? It turns out that you can do that through clever use of kinematics Exploit that decay of neutral K (K1) into two pions is much faster than decay of neutral K (K2) into three pions Mass K0 =498 MeV, Mass π0, π+/- =135 / 140 MeV Therefore K2 must have a longer lifetime thank K1 since small decay phase space t1 = ~0.9 x 10-10 sec t2 = ~5.2 x 10-8 sec (~600 times larger!) Beam of neutral kaons automatically becomes beam of |K2> as all |K1> decay very early on… Pure K2 beam after a while! (all decaying into πππ) ! K1 decay early (into pp) Initial K0 beam

The Cronin & Fitch experiment (1/3) Essential idea: Look for (CP violating) K2  pp decays 20 meters away from K0 production point π0 Decay of K2 into 3 pions Incoming K2 beam J.H. Christenson, J.W. Cronin, V.L. Fitch, R. Turley PRL 13,138 (1964) π+ Vector sum of p(π-),p(π+) π- If you detect two of the three pions of a K2  ppp decay they will generally not point along the beam line

The Cronin & Fitch experiment (2/3) Essential idea: Look for (CP violating) K2  pp decays 20 meters away from K0 production point Decaying pions Incoming K2 beam J.H. Christenson et al., PRL 13,138 (1964) If K2 decays into two pions instead of three both the reconstructed direction should be exactly along the beamline (conservation of momentum in K2  pp decay)

Weak interactions violate CP The Cronin & Fitch experiment (3/3) K2  pp decays (CP Violation!) Weak interactions violate CP Effect is tiny, ~0.05% ! K2  ppp decays K2  p+p-+X p+- = pp+ + pp- q = angle between pK2 and p+- If X = 0, p+- = pK2 : cos q = 1 If X  0, p+-  pK2 : cos q  1 Note scale: 99.99% of K ppp decays are left of plot boundary Result: an excess of events at Q=0 degrees!

Almost but not quite!

with |ε| <<1

Key Points So Far – with term definitions K0, K0 are flavour eigenstates, also known as strong eigenstates These are not CP eigenstates, CP eigenstates are linear combinations of these Short lived and long-lived Kaon states are observed These are physical states with definite lifetimes and masses (mass eigenstates) CP eigenstates (K1, K2) would describe these if CP were conserved CP Violated (a tiny bit) in Kaon decays Physical states (Ks,KL) thus are not (exactly) the CP eigenstates They are eigenstates of the total Hamiltonian (strong + weak) Describe this through Ks, KL as mixture of K0, K0 or K1, K2

Mixing in neutral mesons HEALTH WARNING : We are about to change notation P1,P2 are like Ks, KL (rather than K1,K2)

Particle can transform into its own anti-particle neutral meson states Po, Po P could be Ko, Do, Bo, or Bso Kaon oscillations s d _ W- _ u, c, t u, c, t s W+ d - K0 K0 d s u, c, t _ _ W- W+ s u, c, t d So say at t=0, pure Ko, later a superposition of states

Here for general derivation we have labelled states 1,2

No Mixing – Simplest Case neutral meson states Po, Po P could be Ko, Do, Bo, or Bso 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 (e.g. strangeness)

H is the total hamiltonian: Time evolution of neutral mesons mixed states (1/4) neutral meson states Po, Po P could be Ko, Do, Bo, or Bso 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 H11=H22 from CPT invariance (mass/lifetime particle = antiparticle) (e.g. strangeness) H is the total hamiltonian: EM+strong+weak

Time evolution of neutral mesons mixed states (2/4) Solve Schrödinger for the eigenstates of H : of the form with complex parameters p and q satisfying Time evolution of the eigenstates: Compare with Ks, KL as mixtures of K0, K0 If equal mixtures, like K1 K2

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

Time evolution of neutral mesons mixed states (4/4) Evolution of weak/flavour eigenstates: Time evolution of mixing probabilities: decay terms Interference term i.e. if start with P0, what is probability that after time t that have state P0 ? Parameter x determines “speed” of oscillations compared to the lifetime

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

Summary of Neutral Meson Mixing Lifetimes very different (factor 600) x ~ 0.95 x = 0.00419 ± 0.00211 Δmd = 0.507 ± 0.004 ps−1 xd = 0.770 ± 0.008 Δms = 17.719 ± 0.043 ps−1 xs = 26.63 ± 0.18

Key Points So Far K0, K0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states observed CP Violated (a tiny bit) in Kaon decays Describe this through Ks, KL as mixture of K0 K0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P0 P0 Mass differences and width determine the rates of oscillations Very different for different mesons (Bs,B,D,K)

Cabibbo theory and GIM mechanism

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

Cabibbo rotation and angle (2/3) Cabibbo’s theory successfully correlated many decay rates by counting the number of cosqc and sinqc terms in their decay diagram: E.g.

Cabibbo rotation and angle (3/3) There was however one major exception which Cabibbo could not describe: K0  m+ m- (branching ratio ~7.10-9) Observed rate much lower than expected from Cabibbo’s rate correlations (expected rate  g8 sin2qc cos2qc) d s cosqc sinqc u W W nm m+ m-

The GIM mechanism (1/2) 2D rotation matrix 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 4th quark : the charm (c) quark ! … discovered experimentally in 1974: J/Y  cc state 2D rotation matrix Lepton sector unmixed Quark section mixed through rotation of weak w.r.t. strong eigenstates by qc

The GIM mechanism (2/2) There is also an interesting symmetry between quark generations: u c W+ W+ d’=cos(qc)d+sin(qc)s s’=-sin(qc)d+cos(qc)s 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)

GIM suppression expected rate  (g4 sinqc cosqc - g4 sinqc cosqc)2 The model also explains the smallness of the K0  m+ m- decay See also Bs  m+ m- discussion later d s d s cosqc sinqc -sinqc cosqc u c W W W W nm nm m+ m- m+ m- expected rate  (g4 sinqc cosqc - g4 sinqc cosqc)2 The cancellation is not perfect – these are only the vertex factors – as the masses of c and u are different

The CKM matrix and the Unitarity Triangle

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

The CKM matrix (1/2) u d’ c s’ t b’ 3D rotation matrix 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) This matrix relates the weak states to the mass states Kobayashi Maskawa u d’ c s’ t b’ Imagine a new doublet of quarks 3D rotation matrix 2D rotation matrix

The CKM matrix (2/2) U = u c t D = d s b Can estimate Standard Model weak charged current Feynman diagram amplitude proportional to Vij Ui Dj U (D) are up (down) type quark vectors Vij is the quark mixing matrix, the CKM matrix for 3 families this is a 3x3 matrix U = u c t D = d s b Can estimate relative probabilities of transitions from factors of |Vij |2

CKM matrix – number of parameters (1/2) As the CKM matrix elements are connected to probabilities of transition, the matrix has to be unitary: Values of elements: a purely experimental matter In general, for N generations, N2 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

CKM matrix – number of parameters (2/2) NxN complex element matrix: 2N2 parameters Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases) Number of phases Example for N = 1 generation: 2 unknowns – modulus and phase: unitarity determines |V | = 1 the phase is arbitrary (non-physical) no phase, no CPV

CKM matrix – number of parameters (2/2) NxN complex element matrix: 2N2 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

CKM matrix – number of parameters (2/2) NxN complex element matrix: 2N2 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 3rd family of quarks!

CKM matrix – Particle Data Group (PDG) parameterization 3D rotation matrix form Define: Cij= cos ij Sij=sin ij 3 angles 12, 23, 13 phase  VCKM = R23 x R13 x R12 C12 S12 0 -S12 C12 0 0 0 1 0 0 0 C23 S23 0 -S23 C23 R23 = R12 = C13 0 S13 e-i 0 1 0 -S13 e-i 0 C13 R13 =

CKM matrix - Wolfenstein parameters Introduced in 1983: 3 angles  = S12 , A = S23/S212 ,  = S13cos/ S13S23 1 phase  = S13sin/ S12S23 A ~ 1, ~ 0.22, ≠ 0 but  ≠ 0 ??? VCKM(3) terms in up to 3 CKM terms in 4,5 Note: smallest couplings are complex ( CP-violation)

CKM matrix - Wolfenstein parameters Introduced in 1983: 3 angles  = S12 , A = S23/S212 ,  = S13cos/ S13S23 1 phase  = S13sin/ S12S23 A ~ 1, ~ 0.22, ≠ 0 but  ≠ 0 ??? VCKM(3) terms in up to 3 CKM terms in 4,5 Note: smallest couplings are complex ( CP-violation)

CKM matrix - Wolfenstein parameters Introduced in 1983: 3 angles  = S12 , A = S23/S212 ,  = S13cos/ S13S23 1 phase  = S13sin/ S12S23 ^ ^ Using ρ,η rather than ρ,η adds higher order correction terms A ~ 1, ~ 0.22, ≠ 0 but  ≠ 0 ??? VCKM(3) terms in up to 3 CKM terms in 4,5 Note: smallest couplings are complex ( CP-violation)

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

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 Hence, get a triangle ‘Unitarity’ or ‘CKM triangle’ Triangle if SM is correct. Otherwise triangle will not close, Angles won’t add to 180o Imaginary Real

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

The Unitarity Triangle(s) & the a, b, g angles Area of all the triangles is the same (6A2) 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

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

Key Points So Far K0, K0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states observed CP Violated (a tiny bit) in Kaon decays Describe this through Ks, KL as mixture of K0 K0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P0 P0 Mass differences and width determine the rates of oscillations Very different for different mesons (Bs,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

Types of CP violation We discussed earlier how CP violation can occur in Kaon (or any P0) mixing if p≠q. We didn’t consider the decay of the particle – this leads to two more ways to violate CP

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

1) CP violation in decay (also called direct CP violation) 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 f : complex phase due to weak interactions Valid for both charged and neutral particles P (other types are neutral only since involve oscillations)

2) CP violation in mixing (also called indirect CP violation) Mass eigenstates being different from CP eigenstates Mixing rate for P0  P0 can be different from P0  P0 If CP conserved : If CP violated : with (This is the case if Ks=K1, KL=K2) such asymmetries usually small need to calculate M,, involve hadronic uncertainties hence tricky to relate to CKM parameters

3) CP violation in the interference of mixing and decay Say we have a particle such that P0  f and P0  f are both possible There are then 2 possible decay chains, with or without mixing! Interference term depends on Can put and get but CP can be conserved in mixing and in decay, and still be violated overall !

Key Points So Far Weak and mass eigenstates of quarks are not the same K0, K0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states observed CP Violated (a tiny bit) in Kaon decays Describe this through Ks, KL as mixture of K0 K0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P0 P0 Mass differences and width determine the rates of oscillations Very different for different mesons (Bs,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