Part II
CP Violation in the SM
Chris Parkes
1
Chris Parkes
Outline
THEORETICAL CONCEPTS
I.
Introductory concepts
Matter and antimatter
Symmetries and conservation laws
Discrete symmetries P, C and T
II.
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
Chris Parkes
2
Kaons and
discovery of CP violation
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?
Initially CP appears to
be preserved
in weak
interactions …!
Chris Parkes
Intrinsic
spin

+
+
+
P +

CP

C 

4
Introducing kaons
Kaon mesons: in two isospin doublets
Part of pseudo-scalar JP=0 mesons octet with , h
K+ = us
I3=+1/2
Ko = ds
Ko = ds
I3=-1/2
K- = us
S=+1
S=-1
 Kaon production: (pion beam hitting a target)
Ko : - + p  o + Ko
S
0
0
-1
+1
But from baryon number conservation:
Requires higher
energy
Ko : + + p  K+ + Ko + p
S
Or
0
0
+1
-1
0
Ko : - + p  o + Ko + n +n
S
Chris Parkes
0
0
+1
-1
0
Much higher
0
5
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?
 P K = -1 K
(because l=0 q qbar pair)
 P K = -1 K
(because l=0 q qbar pair)


CK
0
0
0
0
0
= -1 K 0
C K 0 = -1 K 0
effect of CP :
CP K 0 = +1 K
CP K

0
0
= +1 K 0
Bottom line: the flavour eigenstates K0 and K0 are not CP eigenstates
Chris Parkes
6
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 given 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

Chris Parkes
(difference between K2 and KL to be discussed later)
7
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…
K1 decay early (into )
Pure K2 beam after a while!
(all decaying into πππ) !
Initial
K0
beam
Chris Parkes
8
The Cronin & Fitch experiment (1/3)
Essential idea: Look for (CP violating)
K2   decays 20 meters away from
K0 production point
π0
Decay of K2 into 3 pions
Incoming K2 beam
J.H. Christenson, J.W. Cronin,
π+
V.L. Fitch, R. Turley
PRL 13,138 (1964)
Vector sum of p(π-),p(π+)
π-
If you detect two of the three pions
of a K2   decay they will generally
not point along the beam line
Chris Parkes
9
The Cronin & Fitch experiment (2/3)
Essential idea: Look for (CP violating)
K2   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   decay)
Chris Parkes
10
The Cronin & Fitch experiment (3/3)
K2   decays
(CP Violation!)
Weak interactions
violate CP
Effect is tiny, ~0.05% !
K2   decays
Note scale: 99.99% of K  decays
are left of plot boundary
Result: an excess of events at Q=0 degrees!
Chris Parkes
K2  ++X
p+ = p+ + p
q = angle between pK2 and p+
If X = 0, p+ = pK2 : cos q = 1
If X  0, p+  pK2 : cos q  1
11
Chris Parkes
12
K s ¹ K1
KL ¹ K2
Almost but not quite!
Chris Parkes
13
KS =
KL =
2
(K
2
(K
1
1+ e
1
1+ e
1
2
+ e K2
)
+ e K1
)
with |ε| <<1
Chris Parkes
14
p=
Chris Parkes
q=
1
2(1+|e |2 )
1
2(1+|e |2 )
(1+ e )
(1- e )
15
Key Points So Far
• K0, K0 are not CP eigenstates – need to make linear combination
• Short lived and long-lived Kaon states
• CP Violated (a tiny bit) in Kaon decays
• Describe this through Ks, KL as mixture of K0 K0
Chris Parkes
16
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
0
K

d
_
s
d
_
s

u, c, t

W
W
+
W
u, c, t
u, c, t
u, c, t
W+
s
_
0
dK
s
_
d
So say at t=0, pure Ko,
– later a superposition of states
Chris Parkes
18
Chris Parkes
19
Chris Parkes
Here for general derivation we have labelled states 1,2
20
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
 (t )  a (t ) P o + b(t ) P o
 obeys time-dependent Schrödinger equation
æ M
H ºç
è 0
æ a ö
dæ a ö
i
ç
÷ = Hç
÷
dt è b ø
è b ø
0 ö iæ G 0 ö
÷- ç
÷
M ø 2è 0 G ø
 M, : hermitian 2x2 matrices, mass matrix and decay matrix
 mass/lifetime particle = antiparticle
 Solution of form
i
-i (m- G)t
2
y (t) = Ae
Chris Parkes
-imt - G2 t
= Ae
e
21
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
 (t )  a (t ) P o + b(t ) P o
 obeys time-dependent Schrödinger equation
H is the total
hamiltonian:
EM+strong+weak
a
d a
i a
i    H   (M  Γ) 
dt  b 
2 b
b
 M, : hermitian 2x2 matrices, mass matrix and decay matrix
 H11=H22 from CPT invariance (mass/lifetime particle = antiparticle)
 M
H   *
 M 12
Chris Parkes
M 12  i  
   *
M  2  12
12 


22
Time evolution of neutral mesons mixed states (2/4)
 Solve Schrödinger for the eigenstates of H :
 of the form
P1  p P o + q P o
Compare with Ks, KL as mixtures of K0, K0
P2  p P o  q P o
 with complex parameters p and q satisfying
p + q 1
2
2
If equal mixtures, like K1 K2
1
p= q=
2
 Time evolution of the eigenstates:
P1 (t )  P1 e
i
i ( m1  1 ) t
2
P2 (t )  P2 e
Chris Parkes
i
i ( m2  2 ) t
2
23
Time evolution of neutral mesons mixed states (3/4)
e.g. m1, 2  M 
 Some facts and definitions:
i
m i

E1  m1  1  ( M 
)  ( 
)
2
2
2
2
i
m i

E2  m2  2  ( M +
)  ( +
)
2
2
2
2
m
2
m  m2  m1
  2  1
 Characteristic equation
H  EI  0
i
i
i
( M    E ) 2  ( M 12  12 )( M *12  12* )
2
2
2
 Eigenvector equation:
 p 
( H  EI )    0
 q
Chris Parkes
p

q
i
M 12  12
2
i *
*
M 12   12
2
24
Time evolution of neutral mesons mixed states (4/4)
 Evolution of weak/flavour eigenstates:
q
f  (t ) P 0
p
p
+ f  (t ) P 0
q
P 0 (t )  f + (t ) P 0 +
P 0 (t )  f + (t ) P 0
i ( m2  2 ) t 
1  i ( m1  2 1 )t
2
f  (t )  e
e

2

i
i
decay terms
Interference term
2
1 -G1t -G2t
-Gt
é
f± (t) = é
e
+
e
±
2e
cos(Dmt)
é
4é
 Time evolution of mixing probabilities:

i.e. if start with P0, what is probability that after
time t that have state P0 ?
P( P o  P o ;t)  P o P o(t)
P( P  P ;t)  P P (t)
o
Chris Parkes
o
o
o
2
2
 f +(t)
x
2
q

f (t)
p
2
m

1 + 2
2
DG
y=
2G
Parameter x determines
“speed” of oscillations
compared to the lifetime 25
Hints: for proving probabilities
Starting point
æ P1 (0) ö æ p q öæ P0 (0)ö
֍
÷
ç
÷ = çç
ç
è P2 (0)ø è p -q ÷øè P0 (0)÷ø
Turn this around, gives
æ
ç
æ P0 (0)ö ç
ç
÷=ç
ç 0 ÷
è P (0)ø ç
ç
è
1 ö
÷
2 p ÷æ P1 (0) ö
ç
÷
÷
P
(0)
1
1 è 2 ø
÷
2q
2q ÷ø
1
2p
Time evolution
æ P1 (t) ö æ e-i (m1- 2i G1 )t
0
ç
=
ç
÷ ç
-i (m2 - 2i G 2 )t
P
(t)
è 2 ø è
0
e
Use these to find
Chris Parkes
öæ P (0) ö
÷ç 1 ÷
÷è P2 (0)ø
ø
P(P ® P ; t) = P P (t)
o
o
o
o
2
26
Summary of Neutral Meson Mixing
Lifetimes very different
(factor 600)
x = 0.00419 ± 0.00211
x ~ 0.95
Δmd = 0.507 ± 0.004 ps−1
xd = 0.770 ± 0.008
Chris Parkes
Δms = 17.719 ± 0.043 ps−1
xs = 26.63 ± 0.18
27
Key Points So Far
• K0, K0 are not CP eigenstates – need to make linear combination
• Short lived and long-lived Kaon states
• 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)
Chris Parkes
28
Cabibbo theory
and
GIM mechanism
Chris Parkes
29
Cabibbo rotation and angle (1/3)

In 1963 N. Cabibbo made the first step to formally incorporate
strangeness violation in weak decays
1) For the leptons, transitions only occur within a generation
2) For quarks the amount of strangeness violation can be neatly described
in terms of a rotation, where qc=13.1o
Weak
force
transitions
 e     u
  ,   ,  
 e  L    L  d  L
Idea:
weak interaction couples to
different eigenstates than strong
interaction
weak eigenstates can be written
as rotation of strong
eigenstates
Chris Parkes
u

u 
     d cos q + s sin q 
 d L 
c
c L
u
W+
d’ = dcosqc + ssinqc
30
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:
g cos qC
g
E.g.


  n  pe    g cos q
    pe    g sin q
    e  e   g 4

0
Chris Parkes
4
e

purely leptonic
semi-leptonic, S  0
2
C
e
4
g sin qC
2
C
semi-leptonic, S  1
31
Cabibbo rotation and angle (3/3)

There was however one major exception which Cabibbo
could not describe: K0  + - (branching ratio ~7.10-9)
– Observed rate much lower than expected from Cabibbo’s rate
correlations (expected rate  g8 sin2qc cos2qc)
s
d
cosqc
u
sinqc
W
W

+
Chris Parkes
32
The GIM mechanism (1/2)

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
e
 ,  ,
 e    
Lepton
sector
unmixed

2D rotation matrix
u  c
 ,  
 d '   s' 
 d '   cos q c
   
 s'    sin q c
sin q c  d 
 
cos q c  s 
Quark section mixed through
rotation of weak w.r.t. strong
eigenstates by qc
The Cabibbo-GIM model postulates the existence of a 4th quark :
the charm (c) quark !
… discovered experimentally in 1974: J/Y  cc state
Chris Parkes
33
The GIM mechanism (2/2)

There is also an interesting symmetry between quark generations:
u
c
W+
d’=cos(qc)d+sin(qc)s
 d '   cos q c
   
 s '    sin q c
W+
s’=-sin(qc)d+cos(qc)s
sin q c  d 
 
cos q c  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)
Chris Parkes
34
GIM suppression
See also Bs  + discussion later

The model also explains the smallness
of the K0  + - decay
s
d
cosqc
u
sinqc
-sinqc
W
W
c
cosqc
W
W

+
s
d

-
+
-
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
Chris Parkes
35
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:
Pxˆ   xˆ, Ppˆ   pˆ
Txˆ  xˆ, Tpˆ   pˆ
hence “anti-unitary” T (and CP) operation corresponds to complex conjugation !
Since H = H(Vij), complex Vij would generate [T,H]  0  CP violation
A Ui  D j   A Ui  D j 
CP conservation is:
Ui  D j
Ui
Chris Parkes
Ui  D j
W+
=
Vij
Dj
Ui
(up to unphysical phase)
W
only if:
Vij
Vij  Vij
Dj
37
The CKM matrix (1/2)
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:
u
d’
c
s’
2D rotation matrix
 d '   cos q c
   
 s'    sin q c
sin q c  d 
 
cos q c  s 
t
b’
Kobayashi
Maskawa
Imagine a new
doublet of quarks
3D rotation matrix
 d '  Vud Vus Vub  d 
  
 
s
'

V
V
V
   cd
cs
cb  s 
 b'   V
 
   td Vts Vtb  b 
… as mass and flavour eigenstates need not be the same (rotated)
 In other words this matrix relates the weak states to the physical states
Chris Parkes
 d '
d 
 
 
 s '   VCKM  s 
 b' 
b
 
 
38
The CKM matrix (2/2)

Standard Model weak charged current
Feynman diagram amplitude proportional to
•
Ui  D j
U (D) are up (down) type quark vectors
U=

Vij Ui Dj
u
c
t
D=
d
s
b
Vij is the quark mixing matrix, the CKM matrix
•
for 3 families this is a 3x3 matrix
VCKM
Chris Parkes
Vud

  Vcd
V
 td
Vus Vub 

Vcs Vcb 
Vts Vtb 
Ui
Vij
W+
Dj
Can estimate
relative probabilities
of transitions from
factors of |Vij |2
39
CKM matrix – number of parameters (1/2)
 As the CKM matrix elements are connected to probabilities of
transition, the matrix has to be unitary:
*
V
V
 ij jk   ik
j
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
Chris Parkes
40
CKM matrix – number of parameters (2/2)
 NxN complex element matrix: 2N2 parameters
Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases)
2N 2 - N 2 - (2N -1) = N 2 - 2N +1
Number of phases
(N 2 - 2N +1)- N(N -1) / 2 = 12 N(N - 3)+1
Example for N = 1 generation:
i
2 unknowns – modulus and phase: | V | e
unitarity determines |V | = 1
V1gen = (1)
the phase is arbitrary (non-physical)
no phase, no CPV
Chris Parkes
41
CKM matrix – number of parameters (2/2)
 NxN complex element matrix: 2N2 parameters
Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases)
2N 2 - N 2 - (2N -1) = N 2 - 2N +1
Number of phases
(N 2 - 2N +1)- N(N -1) / 2 = 12 N(N - 3)+1
Example for N = 2 generations:
8 unknowns – 4 moduli and 4 phases
unitarity gives 4 constraints :
1
VV †  
0
0
1 
æ V V
ud
us
VCabbibo = ç
ç Vcd Vcs
è
ö æ cosq
sinq c
c
÷=ç
÷ ç -sinq c cosq c
ø è
ö
÷
÷
ø
for 4 quarks, we can adjust 3 relative phases
only one parameter, a rotation (= Cabibbo angle) left: no phase  no CPV
Chris Parkes
42
CKM matrix – number of parameters (2/2)
 NxN complex element matrix: 2N2 parameters
Total - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases)
2N 2 - N 2 - (2N -1) = N 2 - 2N +1
Number of phases
(N 2 - 2N +1)- N(N -1) / 2 = 12 N(N - 3)+1
Example for N = 3 generations:
18 unknowns – 9 moduli and 9 phases
unitarity gives 9 constraints
VCKM
for 6 quarks, we can adjust 5 relative phases
Vud

  Vcd
V
 td
Vus Vub 

Vcs Vcb 
Vts Vtb 
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!
Chris Parkes
43
CKM matrix – Particle Data Group (PDG) parameterization
3D rotation matrix form
 3 angles q12, q23, q13 phase 
Define:
Cij= cos qij
Sij=sin qij
VCKM = R23 x R13 x R12
R23 =
1 0 0
0 C23 S23
0 -S23 C23
R13 =
0
0
1
0
0
C13
S12 0
-S12 C12 0
0
C13
-S13 e-i
Chris Parkes
R12 =
C12
0
1
S13 e-i
44
CKM matrix - Wolfenstein parameters
Introduced in 1983:
 3 angles
 = S12 , A = S23/S212 ,
 = S13cos/ S13S23
 1 phase
h = S13sin/ S12S23
VCKM
æ 1 0 0 ö
ç
÷
» ç 0 1 0 ÷ + O(l ) + O(l 2 ) +...
ç 0 0 1 ÷
è
ø
A ~ 1,  ~ 0.22,  ≠ 0 but h ≠ 0 ???
VCKM(3) terms in up to 3
CKM terms in 4,5
Chris Parkes
 2 
 2 
ˆ   1  , hˆ  h 1  
2
2


Note:
smallest couplings are
complex ( CP-violation)
45
CKM matrix - Wolfenstein parameters
Introduced in 1983:
 3 angles
 = S12 , A = S23/S212 ,
 = S13cos/ S13S23
 1 phase
h = S13sin/ S12S23
VCKM
æ 1 l 0 ö
ç
÷
» ç -l 1 0 ÷ + O(l 2 ) + O(l 3 )
ç 0 0 1 ÷
è
ø
A ~ 1,  ~ 0.22,  ≠ 0 but h ≠ 0 ???
VCKM(3) terms in up to 3
CKM terms in 4,5
Chris Parkes
 2 
 2 
ˆ   1  , hˆ  h 1  
2
2


Note:
smallest couplings are
complex ( CP-violation)
46
CKM matrix - Wolfenstein parameters
Introduced in 1983:
 3 angles
 = S12 , A = S23/S212 ,
 1 phase
h = S13sin/ S12S23
VCKM
 = S13cos/ S13S23

2

1

2

2

2 5
     iA  h
1
2
 3 ˆ
2
ˆ   A  iA4h
A

1



i
h



A ~ 1,  ~ 0.22,  ≠ 0 but h ≠ 0 ???
VCKM(3) terms in up to 3
CKM terms in 4,5
Chris Parkes

A   ih 


A2


1



3
 2 
 2 
ˆ   1  , hˆ  h 1  
2
2


Note:
smallest couplings are
complex ( CP-violation)
47
CKM matrix - hierarchy
VCKM
Vud

  Vcd
V
 td
Vus Vub   O(1) O( ) O(3 ) 

 
2
Vcs Vcb    O( ) O(1) O( ) 


3
2

Vts Vtb   O( ) O( ) O(1) 
Charge: +2/3
Charge: 1/3
top
bottom
charm
strange
up
down
 ~ 0.22
flavour-changing transitions
by weak charged current
(boldness indicates transition
probability  |Vij|)
Chris Parkes
48
CKM – Unitarity Triangle
VudVub * +VcdVcb * +VtdVtb *  0
•Three complex numbers, which sum to zero
•Divide by VcdVcb * 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
Imaginary
•Hence, get a triangle
‘Unitarity’ or ‘CKM triangle’
*
*
VudVub
*
VcdVcb
VtdVtb
*
VcdVcb
•Triangle if SM is correct.
Otherwise triangle will not close,
Angles won’t add to 180o
*
V V
1  cd cb *
VcdVcb
Real
49
Unitarity conditions and triangles
3
V
i 1
ij
2
 1 , j  1,2,3
: no phase info.
*
V
V
 ij jk   ik
j
3
*
V
V
 ij ik  0 , j, k  1,2,3 , j  k
i 1
Plot on Argand diagram: 6 triangles in complex plane
db:
VudVub* + Vcd Vcb* + VtdVtb*  0
sb:
VusVub* + VcsVcb* + VtsVtb*  0
ds:
VudVus* + VcdVcs* + VtdVts*  0
ut:
ct:
VudVtd* + VusVts* + VubVtb*  0
uc:
VudVcd* + VusVcs* + VubVcb*  0
Chris Parkes
VcdVtd* + VcsVts* + VcbVtb*  0
50
The Unitarity Triangle(s) & the a, b, g angles
 Area of all the triangles is the same (6A2h)
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
Chris Parkes
51
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,
Chris Parkes
2008 Nobel prize
52
Key Points So Far
• K0, K0 are not CP eigenstates – need to make linear combination
• Short lived and long-lived Kaon states
• 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
Chris Parkes
53
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
P

f
CP in mixing
P

P
f
P
P
f
P
f
CP in interference between mixing and decay
P
P
Chris Parkes
+
P
P
f
+
f
P
f
P
f
55
1) CP violation in decay (also called direct CP violation)
Occurs when a decay and its CP-conjugate decay
Valid for both charged and
neutral particles P
have a different probability
(other types are neutral only
since involve oscillations)
Decay amplitudes can be written as:
P f
P f
Af  f H P
Af  f H P
Af
Af

i i i i
A
e
 i e
i
 Ae
 i i
i
e
i i
1
i
Two types of phase:

Strong phase: CP conserving, contribution from intermediate states

Weak phase  : complex phase due to weak interactions
Chris Parkes
56
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 :
P1  p P o + q P o
P2  p P o  q P o
with
1
pq
2
CP P1  +1 P1
CP P2  1 P2
(This is the case if Ks=K1, KL=K2)
 If CP violated :
i *
M  12
q
2

1
i
p
M 12  12
2
2
*
12
 such asymmetries usually small
 need to calculate M,,
involve hadronic uncertainties
 hence tricky to relate to CKM parameters
Chris Parkes
57
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
q
1
p
Af
Af
* Not necessary to be CP eigenstate
Chris Parkes
1
q Af
l=
p Af
and get
 1
but
Im( l ) ¹ 0
CP can be conserved in
mixing and in decay, and
still be violated overall !
58
Key Points So Far
•
K0, K0 are not CP eigenstates – need to make linear combination
•
Neutral mesons oscillate from particle to anti-particle
•
Short lived and long-lived Kaon states
•
Can describe neutral meson oscillations through mixture of P0 P0
•
CP Violated (a tiny bit) in Kaon decays
•
Mass differences and width determine the rates of oscillations
•
Describe this through Ks, KL as mixture of K0 K0
•
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
Af
¹1
Af
• Decay
• Mixing
• Interference between decay and mixing
Chris Parkes
2
q
¹1
p
æ
Af ö
q
÷¹0
Imçç l =
÷
p
A
f ø
è
59