:Section 3:

Mixing and CP violation in (mostly) neutral mesons

State evolution in neutral mesons• Neutral meson states Po, Po

– Could be Ko, Do, Bo

• With internal quantum number F– Such that F=0 emag/strong, F0 for Hweak

• Obeys time dependent Schrödinger eqn

M, Hermitian 2x2 matrices, mass matrix and decay matrix,

H11=H22 from CPT (mass/decay particle = anti-particle)

oo PtbPtat )()()(

b

ai

b

aH

b

a

dt

di W )

2( ΓM

• Solve Schrödinger for Eigenstates of Hw

• From characteristic equation

• If E1=M1-i1/2, E2= M2-i2/2

and M=M2-M1, = 2- 1

oo

oo

PqPpP

PqPpP

2

1 Complex coeff. p,q obey |q|2+|p|2=1

0 EIHW)

2)(

2()

2(, *

1212*

12122

iM

iME

iMhence

)2

(2

)2

(

)2

(2

)2

(

2

1

immE

immE

• Eigenvector Eqn

• Eigenstates have time evolution:

• then

0)(

q

pEIHW

12*

12*

1212

2

2

iM

iM

q

p

ti

Mi

ti

Mi

ePtP

ePtP

)2

(

22

)2

(

11

22

11

)(

)(

000

000

)()()(

)()()(

Ptfq

pPtftP

Ptfp

qPtftP

with

ti

Miti

Mieetf

)2

()2

( 2211

2

1)(

Finally probabilities…

• With

• The oscillations depend upon the parameter x the speed of oscillations c.f. lifetime

22

22

P(

P(

(t)fp

q(t)PP;t)PP

(t)f(t)PP;t)PP

oooo

oooo

)cos(24

1)( )(2

21 mteetf tt Where = (1+ 2)/2

Interference term

Mx

Mixing probabilities

x in Bs

not yet measure

d

>19

Probability

of finding

Po from initial pure Po beam

• CP in decay

• CP in mixing

• CP in interference between mixing and decay

Types of CP violation

Pff P

ff P

Pff P

P P P

ff PP P P+ +

1) CP Violation in mixing Indirect CP Violation

• Mass eigenstates being different from CP eigenstates• Mixing rate PoPo is different to PoPo

• If conserved

oo

oo

PqPpP

PqPpP

2

1

2

1qp CP|P1> = +1 |P1>

CP|P2> = -1 |P2> with

1212

*1212

*2

2

2

iM

iM

p

q

•If violated Such asymmetries usually small

Need to calculate M,, involve hadronic uncertainties

Hence, tricky to relate to CKM parameters

2) CP Violation in decaydirect CP Violation

• Two types of phase– Weak phase: due to weak interactions (complex CKM elements)– Strong phase: contribution from intermediate states, CP conserving (same sign in both)

fP

fP

PHfA

PHfA

f

ff is final state common to both decays

1

i

iii

i

iii

f

f

ii

ii

eeA

eeA

A

A

occurs for both charged and neutral states P

3)CP violation in the interference of mixing and

decay• Choose state* f, Pof, Pof• Two possible decay chains, with or w/o mixing

• CP can be conserved in mixing and in decay and still be violated !

*Not necessary to be CP eigenstate

1p

q 1f

f

A

Af

f

A

A

p

q

•Interference term depends on

Can put and get 1 but 1Im

:Section 4:

Neutral Kaon system

Ko Ko system• Kaon mesons in two isospin doublets

K+ = us

Ko = ds

K- = us

Ko = ds

S=+1

S=-1

Part of pseudo-scalar JP=0- meson octet with ,

I3=+1/2

I3=-1/2

• Kaon production

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

Kaon oscillations

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

d

su, c, t W

W_

s

d_u, c, t

ds

u, c, tW W

_ sd_

u, c, t___

K0K0

Ko Decay

• In that case

00

2

001

2

12

1

KKKK

KKKK

L

s

CP=+1

CP=-1

2

1qp

Ksoo

Ks+-

KL+-o

KLooo

CP=+1

CP=-1

95.0

s100014.05301.0=

s1004.017.51

s100008.08934.01

110SL

8

LL

10

SS

Mx

mmm

KS branching fractions: 69%, 31%KL branching fractions: 21%, 13%, 66%

mass eigenstatesKS

KL

show

Assume CP

Time dependent probabilities for the neutral kaon case.

t (1/)

Ko Regeneration

• Start with Pure Ko beam– After time all Ks component decayed

• Introduce slab of material in beam– reactions

pKnK

nKpK

0

0

1) Elastic scatttering

2) Charge exchange

3) Hyperon production

00 pK

•Hence Ko absorbed more strongly

sL

emerge

KffKff

KfKfK

2

1

2

12

1 00

i.e. Ks regenerated

Discovery of CP ViolationK1oo

K1+-

K2+-o

K2ooo

CP=+1

CP=-1

So if KL =K1 CP eigenstate, Observe no two pion component

But if broken get:

212

122

||1

1

||1

1

KKK

KKK

s

L

Where quantifies degree of CP violation

BUT Can one find KL decaying into +-?

KL X

KL

p = p + p

= angle between pKL and p

If X = 0, p = pKL: cos = 1If X 0, p pKL: cos 1

cos

KL

m () < mK

J.H. Christenson et al., PRL 13,138 (1964)

Discovery of CP violation

m () = mK

m () > mK

•Mass and angular spectrum

So CP symmetry is violated in the neutral kaon system.

Mass eigenstates (KS and KL) CP eigenstates

Both KS and KL could decay into +--.

•Experimentally well known:The majority of KS decays into - and KL into --.

015.0

0

N

N Small but with profound implications

In KL

Decay final state at time t

- Spin() = 0L- = 0

Initial state at t = 0

KK

KKpp

0

0

S = 0 S = 0)su(K

)su(K

CP(-

i.e. CP eigenstate

K0 at t = 0 decays into

vsK0 at t = 0 decays into

any difference = CP violation

CPLEAR revisited

•Tag Ko/Ko from charged pion/kaon

K0

K0_

CPLEARR+-(t)andR+-(t)_

CP violation

CPLEARCP asymmetry

A(t) = R+-(t) R+-(t)

R+-(t) R+-(t)_

_

large difference!

•CP violation in mixing

•The two mass eigenstates are not CP eigenstates

Kaons: CP violation in DecayCP violation first through existence of certain decay

modes

SW

LW

SW

LW

KH

KH

KH

KH00

00

00,

~2.3x10-3

If CP violation is only in mixing, i.e. independent of decay

00

So, put channel independent term and channel dependent ’ '2,' 00

S

L2

K

K

N

NHence, by measuring only rates:

00S

00L2

00 K

K

N

N

Re612

2

00get

So, two expts in the 80’s did it:

• NA31 (CERN)• E731 (Fermilab)

• Ambiguous result!

So, two expts did it again…….

3

3

1059.074.0'

Re

1065.03.2'

Re

NA48

KTeV

Measure

and at the same time: L

00LS

00S , NNNN

NA31, NA48

KS is regenerated from KL: SL

00S

00L , rNNrNN

E731, KTeV

No normalization is required,

but efficiencies, acceptances etc. have to be corrected…

Normalisation constants

Effort over30 years!

Note; 3 Re = |/ |i.e.Re 0

Not easy to

compare with SM

theory

CKM parameters with CP conserving parameters

• |Vud| : nuclear Beta decay 0.97340.0008

• |Vus| : semileptonic Kaon and hyperon decay (CCFR) 0.21960.0026

• |Vcd| : Neutrino and anti-neutrino production of charm off valence d quarks0.224 0.016

• |Vcs| : W decays (LEP/me!)0.9960.013

• |Vcb| : semileptonic inclusive and exclusive B decays (LEP/CLEO)0.04120.0020

• |Vub| end point spectrum in semileptonic B decays(LEP/CLEO)0.0036 0.0007

• Bo mixing xd, + lattice gauge inputs |Vtb*Vtd| 0.0079 0.0015• Can use Unitarity constraints