Part II CP Violation in the SM. C hris P arkes. 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
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discovery of CP violation
Weak interactions experimentally proven to:
conserved or violated?
+
Intrinsic
spin
C
P
+
+
Initially CP appears to be preservedin weakinteractions …!
+
CP
Kaon mesons: in two isospin doublets
Part of pseudoscalar JP=0 mesons octet with p, h
Introducing kaons
K+ = us
Ko = ds
K = us
Ko = ds
I3=+1/2
I3=1/2
S=+1
S=1
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
Muchhigher
S 0 0 +1 1 0 0
Now we know the quark contents: K0=sd,K0 =sd
First: what is the effect of C and P on the K0 andK0particles?
(because l=0q qbarpair)
(because l=0q qbarpair)
effect of CP:
Bottom line: the flavoureigenstatesK0 andK0are notCP eigenstates
Neutral kaons (1/2)
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 (“Klong”) was discovered in 1956 after being predicted
(difference between K2 and KL to be discussed later)
Neutral kaons (2/2)
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 1010 sec
t2= ~5.2 x 108 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
Pure K2 beam after a while!(all decaying into πππ) !
K1 decay early (into pp)
Initial K0beam
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 pionsof a K2 ppp decay they will generallynot 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 ofthree both the reconstructed directionshould be exactly along the beamline(conservation of momentum in K2 pp decay)
The Cronin & Fitch experiment (3/3)
K2 ppdecays(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 decaysare left of plot boundary
Result: an excess of events at Q=0 degrees!
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 antiparticle
Kaon oscillations
s
d
W
_
_
u, c, t
u, c, t
s
W+

d
K0
K0
d
s
u, c, t
_
_
W+
W
u, c, t
s
d
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 timedependent Schrödinger equation
M, : hermitian 2x2 matrices, mass matrix and decay matrix
mass/lifetime particle = antiparticle
Solution of form
No Mixing – Simplest Case
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 timedependent Schrödinger equation
M, : hermitian 2x2 matrices, mass matrix and decay matrix
H11=H22 from CPT invariance (mass/lifetime particle = antiparticle)
Time evolution of neutral mesons mixed states (1/4)
H is the total hamiltonian:
EM+strong+weak
Solve Schrödinger for the eigenstates of H:
of the form
with complex parameters p and q satisfying
Time evolution of the eigenstates:
Time evolution of neutral mesons mixed states (2/4)
Compare with Ks, KL as mixtures of K0, K0
If equal mixtures, like K1 K2
Time evolution of neutral mesons mixed states (3/4)
e.g.
Time evolution of neutral mesons mixed states (4/4)
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
xd = 0.770 ± 0.008
Δms= 17.719 ± 0.043 ps−1
xs = 26.63 ± 0.18
Lifetimes very different (factor 600)
x = 0.00419 ± 0.00211
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
Cabibbo rotation and angle (1/3)
Weakforcetransitions
u
Idea: weak interaction couples to different eigenstates than strong interactionweak eigenstates can be writtenas rotation of strongeigenstates
W+
d’ = dcosqc + ssinqc
Cabibbo’s theory successfully correlated many decay rates by counting the number ofcosqcandsinqcterms in their decay diagram:
Cabibbo rotation and angle (2/3)
E.g.
There was however one major exception which Cabibbocould not describe: K0 m+m(branching ratio ~7.109)
Observed rate much lower than expected from Cabibbo’s ratecorrelations (expected rate g8 sin2qc cos2qc)
Cabibbo rotation and angle (3/3)
s
d
cosqc
sinqc
u
W
W
nm
m
m+
the uptype quarks weakly decay to “rotated” downtype quarks
the charm (c) quark !
… discovered experimentally in 1974: J/Y cc state
2D rotation matrix
Leptonsectorunmixed
Quark section mixed throughrotation of weak w.r.t. strong eigenstates by qc
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 weakeigenstate d’,
is not the same as the masseigenstate (the d)
See alsoBsm+mdiscussion later
s
d
s
d
cosqc
sinqc
sinqc
cosqc
u
c
W
W
W
W
nm
nm
m
m+
m
m+
expected rate (g4sinqccosqc g4 sinqccosqc)2
The cancellation is not perfect – these are only the vertex factors – as the masses of c and u are different
Unitarity Triangle
Since H = H(Vij), complex Vij would generate [T,H] 0 CP violation
only if:
How to incorporate CP violation in the SM?
Simple exercise:
hence “antiunitary” T (and CP) operation corresponds to complex conjugation !
CP conservation is: (up to unphysical phase)
=
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
The CKM matrix (1/2)
Kobayashi
Maskawa
ud’
c s’
t b’
Imagine a new
doublet of quarks
3D rotation matrix
2D rotation matrix
Standard Model weak charged current
Feynman diagram amplitude proportional to VijUiDj
U (D) are up (down) type quark vectors
Vijis the quark mixing matrix, the CKM matrix
for 3 families this is a 3x3 matrix
u
c
t
d
s
b
U =
D =
The CKM matrix (2/2)
Can estimate
relative probabilities
of transitions from
factors of Vij 2
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 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
2N1 phases are irrelevant
(choose i and j, i≠j)
CKM matrix – number of parameters (2/2)
Total  unitarity constraints  phase freedom: ‘free’ parameters (rotations +phases)
Number of phases
no phase, no CPV
CKM matrix – number of parameters (2/2)
Total  unitarity constraints  phase freedom: ‘free’ parameters (rotations +phases)
Number of phases
only one parameter, a rotation (= Cabibbo angle) left: no phase no CPV
CKM matrix – number of parameters (2/2)
Total  unitarity constraints  phase freedom: ‘free’ parameters (rotations +phases)
Number of phases
4 unknown parameters left: 3 rotation (Euler) angles and 1 phase CPV !
In requiring CP violation with this structureof weak interactions K&M predicteda 3rd family of quarks!
CKM matrix – Particle Data Group (PDG) parameterization
3D rotation matrix form
Define:
Cij= cos ij
Sij=sin ij
VCKM = R23 x R13 x R12
C12 S12 0
S12 C12 0
0 0 1
R23 =
R12 =
C13 0 S13 ei
0 1 0
S13ei 0 C13
R13 =
CKM matrix  Wolfenstein parameters
Introduced in 1983:
= S12 , A = S23/S212 , = S13cos/ S13S23
= S13sin/ S12S23
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
VCKM(3) terms in up to 3
CKMterms in4,5
Note:smallest couplings are complex (CPviolation)
CKM matrix  Wolfenstein parameters
Introduced in 1983:
= S12 , A = S23/S212 , = S13cos/ S13S23
= S13sin/ S12S23
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
VCKM(3) terms in up to 3
CKMterms in4,5
Note:smallest couplings are complex (CPviolation)
CKM matrix  Wolfenstein parameters
Introduced in 1983:
= S12 , A = S23/S212 , = S13cos/ S13S23
= S13sin/ S12S23
A ~ 1, ~ 0.22, ≠ 0 but ≠ 0 ???
VCKM(3) terms in up to 3
CKMterms in4,5
Note:smallest couplings are complex (CPviolation)
Charge: +2/3
Charge: 1/3
~ 0.22
top
bottom
strange
charm
flavourchanging transitions by weak charged current (boldness indicates transition probability Vij)
up
down
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
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
CP in mixing
CP in interference between mixing and decay
Types of CP violation
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 CPconjugate decay
have a different probability
Decay amplitudes can be written as:
Two types of phase:
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)
with
(This is the case if Ks=K1, KL=K2)
involve hadronic uncertainties
3) CP violation in the interference of mixing and decay
P0 f and P0 f are both possible
CP can be conserved in mixing and in decay, and still be violated overall !
* Not necessary to be CP eigenstate