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1Chris Parkes
Part II
CP Violation in the SM
Chris Parkes
2Chris Parkes
Outline
THEORETICAL CONCEPTS
I. Introductory conceptsMatter and antimatter
Symmetries and conservation laws
Discrete symmetries P, C and T
II. CP Violation in the Standard ModelKaons 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
Kaons and
discovery of CP violation
4Chris Parkes
What about the product CP?
+
+
+
+
Intrinsicspin
P C
CP
Initially CP appears to be preservedin weakinteractions …!
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?
5Chris Parkes
Kaon mesons: in two isospin doublets
Part of pseudo-scalar JP=0- mesons octet with p, h
Introducing kaons
K+ = us
Ko = dsKo = ds
K- = us
S=+1 S=-1
I3=+1/2
I3=-1/2
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
Much higher
S 0 0 -1 +1
S 0 0 +1 -1 0
S 0 0 +1 -1 0 0
6Chris Parkes
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)
(because l=0 q qbar pair)
effect of CP :
Bottom line: the flavour eigenstates K0 and K0 are not CP eigenstates
Neutral kaons (1/2)
7Chris Parkes
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
(difference between K2 and KL to be discussed later)
Neutral kaons (2/2)
8Chris Parkes
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…
Looking closer at KL decays
Initial K0
beam
K1 decay early (into pp) Pure K2 beam after a while!(all decaying into πππ) !
9Chris Parkes
Incoming K2 beam
Decay of K2 into 3 pions
If you detect two of the three pionsof a K2 ppp decay they will generallynot point along the beam line
Essential idea: Look for (CP violating) K2 pp decays 20 meters away from K0 production point
The Cronin & Fitch experiment (1/3)
J.H. Christenson, J.W. Cronin,
V.L. Fitch, R. Turley
PRL 13,138 (1964)
π0
π+
π-
Vector sum of p(π-),p(π+)
10Chris Parkes
Incoming K2 beam
Decaying pions
If K2 decays into two pions instead ofthree both the reconstructed directionshould be exactly along the beamline(conservation of momentum in K2 pp decay)
The Cronin & Fitch experiment (2/3)
J.H. Christenson et al.,
PRL 13,138 (1964)
Essential idea: Look for (CP violating) K2 pp decays 20 meters away from K0 production point
11Chris Parkes
Result: an excess of events at Q=0 degrees!
K2 pp decays(CP Violation!)
K2 ppp decays
Note scale: 99.99% of K ppp decaysare left of plot boundary
The Cronin & Fitch experiment (3/3)
K2 p+p-+Xp+- = p +p + p -p
q = angle between pK2 and p+-If X = 0, p+- = pK2 : cos q = 1If X 0, p+- pK2 : cos q 1
Weak interactions violate CP
Effect is tiny, ~0.05% !
Weak interactions violate CP
Effect is tiny, ~0.05% !
12Chris Parkes
13Chris Parkes
Almost but not quite!
14Chris Parkes
with |ε| <<1
15Chris Parkes
16Chris Parkes
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
Mixing
in neutral mesons
HEALTH WARNING :We are about to change notation
P1,P2 are like Ks, KL (rather than K1,K2)
18Chris Parkes
Particle can transform into its own anti-particle
neutral meson states Po, Po
P could be Ko, Do, Bo, or Bso
Kaon oscillations
So say at t=0, pure Ko, – later a superposition of states
ds
u, c, t W-
W+_s
d_u, c, t
d
s u, c, tW- W+_
s
d
u, c, tK0
K0 -
_
19Chris Parkes
20Chris Parkes Here for general derivation we have labelled states 1,2
21Chris Parkes
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 F0 for weak interactions
obeys time-dependent Schrödinger equation
M, : hermitian 2x2 matrices, mass matrix and decay matrix
mass/lifetime particle = antiparticle
Solution of form
oo PtbPtat )()()(
No Mixing – Simplest Case
22Chris Parkes
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 F0 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)
oo PtbPtat )()()(
b
ai
b
a
b
a
dt
di )
2( ΓMH
Time evolution of neutral mesons mixed states (1/4)
*
12
12*12
12
2
i
MM
MMH
H is the total hamiltonian:
EM+strong+weak
23Chris Parkes
Solve Schrödinger for the eigenstates of H :
of the form
with complex parameters p and q satisfying
Time evolution of the eigenstates:
oo
oo
PqPpP
PqPpP
2
1
Time evolution of neutral mesons mixed states (2/4)
122 qp
ti
mi
ti
mi
ePtP
ePtP
)2
(
22
)2
(
11
22
11
)(
)(
Compare with Ks, KL as mixtures of K0, K0
If equal mixtures, like K1 K2
24Chris Parkes
Some facts and definitions:
Characteristic equation
Eigenvector equation:
0)(
q
pEIH
12*
12*
1212
2
2
iM
iM
q
p
Time evolution of neutral mesons mixed states (3/4)
0 EIH )2
)(2
()2
( *1212
*1212
2 i
Mi
MEi
M
12
12
mmm
)2
(2
)2
(2
)2
(2
)2
(2
222
111
imM
imE
imM
imE
22,1
mMm
e.g.
25Chris Parkes
Evolution of weak/flavour eigenstates:
Time evolution of mixing probabilities:
22
22
P(
P(
(t)fp
q(t)PP;t)PP
(t)f(t)PP;t)PP
oooo
oooo
Interference term
mx
000
000
)()()(
)()()(
Ptfq
pPtftP
Ptfp
qPtftP
ti
miti
mieetf
)2
()2
( 2211
2
1)(
Time evolution of neutral mesons mixed states (4/4)
221
i.e. if start with P0, what is probability that after
time t that have state P0 ?
decay terms
Parameter x determines “speed” of oscillations
compared to the lifetime
26Chris Parkes
Hints: for proving probabilitiesStarting point
Turn this around, gives
Time evolution
Use these to find
27Chris Parkes
Δmd = 0.507 ± 0.004 ps−1
xd = 0.770 ± 0.008
Δms = 17.719 ± 0.043 ps−1
xs = 26.63 ± 0.18
x = 0.00419 ± 0.00211
Lifetimes very different (factor 600)
28Chris Parkes
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)
29Chris Parkes
Cabibbo theory
and
GIM mechanism
30Chris Parkes
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
Cabibbo rotation and angle (1/3)
, ,e LL L
e u
d
cos sinc cL L
uu
d sd
Weakforce
transitions
u
d’ = dcosqc + ssinqc
W+
Idea: weak interaction couples to different eigenstates than strong interaction
weak eigenstates can be writtenas rotation of strongeigenstates
31Chris Parkes
Cabibbo’s theory successfully correlated many decay rates by counting the number of cosqc and sinqc terms in their decay diagram:
Cabibbo rotation and angle (2/3)
4
4 2
0 4 2
cos 0
sin 1
purely leptonic
semi-leptonic,
semi-leptonic,
e
e C
e C
e g
n pe g S
pe g S
cos Cg g sin Cg
E.g.
32Chris Parkes
There was however one major exception which Cabibbocould not describe: K0 m+ m- (branching ratio ~7.10-9)
Observed rate much lower than expected from Cabibbo’s ratecorrelations (expected rate g8
sin2qc cos2qc)
Cabibbo rotation and angle (3/3)
d
m+ m-
nm
ucosqc sinqc
WW
s
33Chris Parkes
The GIM mechanism (1/2)
In 1970 Glashow, Iliopoulos and Maiani publish a model forweak 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
'
,'
,,s
c
d
ue
e
s
d
s
d
cc
cc
cossin
sincos
'
'
Leptonsector
unmixed
Quark section mixed throughrotation of weak w.r.t. strong eigenstates by qc
2D rotation matrix
34Chris Parkes
The GIM mechanism (2/2)
There is also an interesting symmetry between quark generations:
u
d’=cos(qc)d+sin(qc)s
W+
c
s’=-sin(qc)d+cos(qc)s
W+
s
d
s
d
cc
cc
cossin
sincos
'
'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)
35Chris Parkes
GIM suppression
d
m+ m-
nm
ucosqc sinqc
WW
s d
m+ m-
nm
ccosqc-sinqc
WW
s
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
See also Bs m+ m- discussion later
The model also explains the smallness
of the K0 m+ m- decay
The CKM matrix and the
Unitarity Triangle
37Chris Parkes
How to incorporate CP violation in the SM?
hence “anti-unitary” T (and CP) operation corresponds to complex conjugation !
Simple exercise:
Since H = H(Vij), complex Vij would generate [T,H] 0 CP violation
W
jDiU
i jU D
ijV
i j i jA U D A U D
W
jDiU
i jU D
ijV = only if:
ij ijV V
How does CP conjugation (or, equivalently, T conjugation)act on the Hamiltonian H ?
CP conservation is: (up to unphysical phase)
Recall:
ˆ ˆˆ ˆ,
ˆ ˆˆ ˆ,
Px x Pp p
Tx x Tp p
38Chris Parkes
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)
In other words this matrix relates the weak states to the physical states
ud’
c s’
t b’
s
d
s
d
cc
cc
cossin
sincos
'
'
b
s
d
VVV
VVV
VVV
b
s
d
tbtstd
cbcscd
ubusud
'
'
'
The CKM matrix (1/2)
Kobayashi Maskawa
Imagine a newdoublet of quarks
b
s
d
V
b
s
d
CKM
'
'
'
2D rotation matrix3D rotation matrix
39Chris Parkes
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 =
uct
D =
dsb
The CKM matrix (2/2)
W
jDiU
i jU D
ijV
tbtstd
cbcscd
ubusud
CKM
VVV
VVV
VVV
V
Can estimate
relative probabilities
of transitions from
factors of |Vij |2
40Chris Parkes
As the CKM matrix elements are connected to probabilities of transition, the matrix has to be unitary:
CKM matrix – number of parameters (1/2)
Values of elements:a purely experimental matterikjk
jijVV *
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
41Chris Parkes
CKM matrix – number of parameters (2/2)
Example for N = 1 generation:
2 unknowns – modulus and phase:
unitarity determines |V | = 1
the phase is arbitrary (non-physical)
| | iV e
no phase, no CPV
NxN complex element matrix: 2N2 parametersTotal - unitarity constraints - phase freedom: ‘free’ parameters (rotations +phases)
Number of phases
42Chris Parkes
CKM matrix – number of parameters (2/2)
NxN complex element matrix: 2N2 parametersTotal - 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
† 1 0
0 1VV
only one parameter, a rotation (= Cabibbo angle) left: no phase no CPV
43Chris Parkes
CKM matrix – number of parameters (2/2)
NxN complex element matrix: 2N2 parametersTotal - 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 structureof weak interactions K&M predicted
a 3rd family of quarks!
tbtstd
cbcscd
ubusud
CKM
VVV
VVV
VVV
V
44Chris Parkes
C12 S12 0
-S12 C12 0
0 0 1
1 0 00 C23 S23
0 -S23 C23
3 angles 12, 23, 13 phase
VCKM = R23 x R13 x R12
R12 =R23 =
R13 =
C13 0 S13 e-i
0 1 0
-S13 e-i 0 C13
CKM matrix – Particle Data Group (PDG) parameterization
Define:
Cij= cos ij
Sij=sin ij
3D rotation matrix form
45Chris Parkes
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
21ˆ ,
21ˆ
22
Introduced in 1983:
3 angles
= S12 , A = S23/S212 , = S13cos/ S13S23
1 phase
= S13sin/ S12S23
VCKM(3) terms in up to 3
CKM terms in 4,5
CKM matrix - Wolfenstein parameters
Note:smallest couplings are complex ( CP-violation)
46Chris Parkes
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
21ˆ ,
21ˆ
22
Introduced in 1983:
3 angles
= S12 , A = S23/S212 , = S13cos/ S13S23
1 phase
= S13sin/ S12S23
VCKM(3) terms in up to 3
CKM terms in 4,5
CKM matrix - Wolfenstein parameters
Note:smallest couplings are complex ( CP-violation)
47Chris Parkes
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
21ˆ ,
21ˆ
22
Introduced in 1983:
3 angles
= S12 , A = S23/S212 , = S13cos/ S13S23
1 phase
= S13sin/ S12S23
VCKM(3) terms in up to 3
CKM terms in 4,5
CKM matrix - Wolfenstein parameters
1ˆˆ12
1
21
423
22
52
32
iAAiA
AiA
iA
VCKM
Note:smallest couplings are complex ( CP-violation)
48Chris Parkes
CKM matrix - hierarchy
)1()()(
)()1()(
)()()1(
23
2
3
OOO
OOO
OOO
VVV
VVV
VVV
V
tbtstd
cbcscd
ubusud
CKM
~ 0.22
top
charm
up down
strange
bottom
Charge: +2/3 Charge: 1/3
flavour-changing transitions by weak charged current (boldness indicates transition probability |Vij|)
49
CKM – Unitarity Triangle
*cbcdVV
0*** tbtdcbcdubud VVVVVV
*
*
cbcd
ubud
VV
VV
• 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
Real
Imag
inar
y
• Hence, get a triangle
‘Unitarity’ or ‘CKM triangle’
• Triangle if SM is correct.
Otherwise triangle will not close,
Angles won’t add to 180o
*
*
1cbcd
cbcd
VV
VV
*
*
cbcd
tbtd
VV
VV
50Chris Parkes
ikjkj
ijVV *
Plot on Argand diagram: 6 triangles in complex plane
3,2,1,123
1
jVi
ij : no phase info.
kjkjVVi
ikij
,3,2,1,,03
1
*
0
0
0
0
0
0
***
***
***
***
***
***
cbubcsuscdud
tbcbtscstdcd
tbubtsustdud
tstdcscdusud
tbtscbcsubus
tbtdcbcdubud
VVVVVV
VVVVVV
VVVVVV
VVVVVV
VVVVVV
VVVVVVdb:
sb:
ds:
ut:
ct:
uc:
Unitarity conditions and triangles
51Chris Parkes
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
52Chris Parkes
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
53Chris Parkes
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
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
55Chris Parkes
CP in decay
CP in mixing
CP in interference between mixing and decay
Pff
P
ff P
Pff P
P P P
ff PP P P
+ +
Types of CP violation
56Chris Parkes
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
fP
fP
PHfA
PHfA
f
f
1
i
iii
i
iii
f
f
ii
ii
eeA
eeA
A
A
1) CP violation in decay (also called direct CP violation)
Valid for both charged and neutral particles P
(other types are neutral only since involve oscillations)
57Chris Parkes
Mass eigenstates being different from CP eigenstates
Mixing rate for P0 P0 can be different from P0 P0
If CP conserved :
If CP violated :
oo
oo
PqPpP
PqPpP
2
1
2
1qpwith
1
2
2
1212
*1212
*2
iM
iM
p
q such asymmetries usually small
need to calculate M,,
involve hadronic uncertainties
hence tricky to relate to CKM parameters
2) CP violation in mixing (also called indirect CP violation)
22
11
1
1
PPCP
PPCP
(This is the case if Ks=K1, KL=K2)
58Chris Parkes
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
* Not necessary to be CP eigenstate
1p
q1
f
f
A
A1
3) CP violation in the interference of mixing and decay
CP can be conserved in mixing and in decay, and still be violated overall !
59Chris Parkes
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
• Three ways for CP violation to occur
• Decay
• Mixing
• Interference between decay and mixing