1. Part II
CP Violation in the SM
Chris Parkes

2. 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

3. discovery of CP violation
Kaons and
discovery of CP violation

4.  +  + +  +   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

5. 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 S
Much higher S

6. 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)

7. 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)

8. 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 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

9. 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

10. 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)

11. 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!

13. Almost but not quite!

14. with |ε| <<1

16. 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

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

18. 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

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

21. 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)

22. 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

23. 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

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

25. 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

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

27. Summary of Neutral Meson Mixing
Lifetimes very different (factor 600)
x ~ 0.95
x = ±
Δmd = ± ps−1
xd = ± 0.008
Δms = ± ps−1
xs = ± 0.18

28. 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)

29. Cabibbo theory
and
GIM mechanism

30. 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

31. 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.

32. 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-

33. 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

34. 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)

35. 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

36. The CKM matrix and the
Unitarity Triangle

37. 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: =

38. 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

39. 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

40. 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

41. 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

42. 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

43. 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!

44. 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
0 C23 S23
0 -S23 C23
R23 =
R12 =
C S13 e-i
-S13 e-i C13
R13 =

45. CKM matrix - Wolfenstein parameters
Introduced in 1983:
3 angles
 = S12 , A = S23/S ,  = 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)

46. CKM matrix - Wolfenstein parameters
Introduced in 1983:
3 angles
 = S12 , A = S23/S ,  = 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)

47. CKM matrix - Wolfenstein parameters
Introduced in 1983:
3 angles
 = S12 , A = S23/S ,  = 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)

48. 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

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
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

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

51. 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

52. 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. 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

54. 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

55. 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

56. 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)

57. 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

58. 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 !

59. 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