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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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slide1

Part II

CP Violation in the SM

Chris Parkes

slide2

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
slide3

Kaons and

discovery of CP violation

slide4

Whatabout 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

C

P



+

+

Initially CP appears to be preservedin weakinteractions …!



+



CP

slide5

Kaon mesons: in two isospin doublets

Part of pseudo-scalar 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

  • 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

Muchhigher

S 0 0 +1 -1 0 0

slide6

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 andK0particles?

(because l=0q qbarpair)

(because l=0q qbarpair)

effect of CP:

Bottom line: the flavoureigenstatesK0 andK0are notCP eigenstates

Neutral kaons (1/2)

slide7

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)

slide8

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

Pure K2 beam after a while!(all decaying into πππ) !

K1 decay early (into pp)

Initial K0beam

slide9

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

slide10

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)

slide11

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!

slide16

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
slide17

Mixing

in neutral mesons

HEALTH WARNING :

We are about to change notation

P1,P2 are like Ks, KL (rather than K1,K2)

slide18

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-

u, c, t

s

d

  • So say at t=0, pure Ko,
    • later a superposition of states
slide21

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

No Mixing – Simplest Case

slide22

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)

Time evolution of neutral mesons mixed states (1/4)

H is the total hamiltonian:

EM+strong+weak

slide23

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

slide24

Time evolution of neutral mesons mixed states (3/4)

  • Some facts and definitions:
  • Characteristic equation
  • Eigenvector equation:

e.g.

slide25

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

slide26

Hints: for proving probabilities

Starting point

Turn this around, gives

Time evolution

Use these to find

slide27

Δmd = 0.507 ± 0.004 ps−1

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

slide28

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)
slide29

Cabibbo theory

and

GIM mechanism

slide30

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’ = dcosqc + ssinqc

slide31

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.

slide32

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)

s

d

cosqc

sinqc

u

W

W

nm

m-

m+

slide33

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

2D rotation matrix

Leptonsectorunmixed

Quark section mixed throughrotation of weak w.r.t. strong eigenstates by qc

slide34

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 weakeigenstate d’,

is not the same as the masseigenstate (the d)

slide35

GIM suppression

  • The model also explains the smallness
  • of the K0 m+m-decay

See alsoBsm+m-discussion 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

slide36

The CKM matrix and the

Unitarity Triangle

slide37

Recall:

Since H = H(Vij), complex Vij would generate [T,H]  0  CP violation

only if:

How to incorporate CP violation in the SM?

  • How does CP conjugation (or, equivalently, T conjugation)act on the Hamiltonian H ?

Simple exercise:

hence “anti-unitary” T (and CP) operation corresponds to complex conjugation !

CP conservation is: (up to unphysical phase)

=

slide38

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

slide39

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

slide40

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

  • 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
slide41

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

slide42

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

slide43

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 structureof weak interactions K&M predicteda 3rd family of quarks!

slide44

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

-S13e-i 0 C13

R13 =

slide45

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

CKMterms in4,5

Note:smallest couplings are complex (CP-violation)

slide46

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

CKMterms in4,5

Note:smallest couplings are complex (CP-violation)

slide47

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

CKMterms in4,5

Note:smallest couplings are complex (CP-violation)

slide48

CKM matrix - hierarchy

Charge: +2/3

Charge: 1/3

~ 0.22

top

bottom

strange

charm

flavour-changing transitions by weak charged current (boldness indicates transition probability  |Vij|)

up

down

ckm unitarity triangle
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

slide50

Unitarity conditions and triangles

: no phase info.

Plot on Argand diagram: 6 triangles in complex plane

db:

sb:

ds:

ut:

ct:

uc:

slide51

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
slide52

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
slide53

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
slide54

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

slide55

CP in decay

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

slide56

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)

slide57

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
slide58

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 !

* Not necessary to be CP eigenstate

slide59

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