D0 -D0 Mixing

Mao-Zhi Yang (杨茂志 )

Jul. 23, 2008, KITPC

Department of PhysicsNankai University

Program on Flavor Physics

Contents

I. Introduction

II. The basic formulas

IV Decays of physical neutral D system

VI Summary

III. The mixing parameters in the SM

V Time-dependent measurement of D0D0 mixing

and can transform into each under weak interaction0D 0D

c

u

u

c

bsd ,,

udV

*cdV udV

*cdV

0D 0D

0D 0D

and can not be separated absolutely0D 0D

I Introduction

The D0-D0 mixing occurs via loop diagrams involving intermediate down-type quarks, it provides unique information about weak interaction

In the standard model, the mixing amplitude is quite small

It is severely suppressed by the GIM mechanisms

)())(,( *

,,,

*cjuj

bsdjiciuiji VVVVmmfA

Loop-integration function

The b-quark contribution is highly suppressed by the CKM factor

)(

1)1(

2/1

)(2/14

23

22

32

O

AiA

A

iA

VCKM

d s b

uc

t

)103(||/|| 4** OVVVV csuscbub

The CKM suppression factor

The b-quark contribution in the loop diagram can be neglected

The mixing amplitude vanishes in the limit of SU(3) flavor symmetry , ms=md, due to the GIM suppression.

Mixing is only the effect of SU(3) breaking

Thus, the mixing in D system involves only the first two generations. CP violation is absent in both the mixing and decay amplitudes, and therefore can be neglected.

breaking])3([sin~ 2 SUT Cmixing

In general the neutral D meson exists as a mixture state of D0 and D0

00 DbDaD

Assume there is a neutral D state at t=0:

00 )0()0()0( DbDa

Then at any time t, the state evolves into

221100 )()()()()( ftcftcDtbDtat

Oscillation within neutral D state

States D decays into

II The basic formulas

00 )()()( DtbDtatD

If we only consider the oscillation within the neutral D state, then we can consider the evolution of the following state

which can be written in the form of matrix product

)(

)()( 00

tb

taDDtD

then we can use

)(

)(

tb

tato stand for the wave

function of the neutral D meson state

The Shrödinger equation for the evolution of the wave function is

)(

)(

)(

)(

tb

taH

tb

ta

ti

22

H needs not be Hermite because D meson can decay in the evolution

2

iMH

MM

22

HHHHH

M 2

i

The matrix H expressed explicitly in term of the matrix elements

22222121

12121111

22

22i

Mi

M

iM

iM

H

The matrix elements are determined by the Hamiltonians of strong, electromagnetic and weak interactions

wemsttotal HHHH

The magnitude of weak interaction is greatly smaller than the strong and electromagnetic interaction

emstw HHH

The weak interaction can be treated as a perturbation over the strong and EM interaction.

Then the matrix elements can be solved perturbatively

2

iM

)(

2

1

2

1

23

112

w

n nD

cw

cw

D

cw

D

HO

iEE

HnnH

mH

m

iM

, ~ 00 , DD

The eigenstates of Hst+Hem

n ~ ,,,,,3,2 eKKK

Using the formula

))(()(

1

)(

1xfi

xfP

ixf

)(11

nDnDnD

EEiEE

PiEE

n nD

cw

cw

D

cw

D iEE

HnnH

mH

m

iM

112

2

1

2

1

2

Then from

n nD

cw

cw

D

cw

D EE

HnnHP

mH

mM

112

2

1

2

1

)(2

1

2

1 11nD

n

cw

cw

D

EEHnnHm

① If CPT is conserved, then M11=M22, and Γ11=Γ22.

Theorems:

② If T is conserved, then Γ12

* M12*

Γ12 M12

The proof can be performed with the formulas in the previous page

The eigen-equation

b

a

b

a

iM

iM

iM

iM

22222121

12121111

22

22

Solve the equation, one can get the eigenvalues and eigenfunctions

])2

)(2

(4)2

(

)(2

[2

1

212112122

221122112,1

iM

iM

im

iMM

2211 MMm 2211

The real parts of will be the masses m1 and m2 of the two eigenstates

2,1

The imaginary parts will be decay widths of the two eigenstates: 21 and,

2,12,12,1 2

im

That is

The two relevant eigenstates are

002

001

11

11

DzqDzpD

DzqDzpD

222 2

im

111 2

im

1212

2121

2

2

iM

iM

p

q

)2

)(2

(4)2

(

)(2

212112122

22112211

iM

iM

im

iMM

z

If CPT is conserved, then z=0

Then

002

001

DqDpD

DqDpD

p and q satisfy the normalization condition

122 qp

III The estimation of the mixing parameters in the SM

Two physical parameters that characterize the mixing are

2212

12

y

mmmx

221

Where Γis the average decay widths of the two eigenstates D1 and D2

Using the eigenvalues of D1 and D2, we can obtain with CPT conservation and neglecting CP violation in mixing

n nD

cw

cw

D

cw

D EE

DHnnHDP

mDHD

m

Mm0110

020

12

11

2

)(21

2 011012 nD

n

cw

cw

D

EEDHnnHDm

where is implicitly included)()2( 33nD pp

Define the correlator

01104 )}0()({ DHxHTDdx cw

cw

Insert a complete set of final states n

nn

One can obtain

)()2(

)}(22

{

33

0110

nD

nDn nDn

cw

cw

pp

EEEE

iPDHnnHD

Compare the above result with the expression of

,m

One can obtain

])}0()({1

Re[2

1

1

0110

020

DHxHTDdxim

DHDm

m

cw

cw

D

cw

D

})}0()({Im[1 0110 DHxHTDdxi

mc

wc

wD

The contribution of the box diagrams

2cwH

The box contribution to is given bym 0~dm

Numerically, by simply taking

which leads to

5box 106.1 x

The bare quark loop contribution to is even further suppressed by additional powers of cs mm /

Numerically, one finds

-7box 10few~ y

The small result of box diagram can be enhanced by various long-distance effects, or by contributions of higher-dimension operator in the OPE

Long-distance effects

0D 0DK

.,,,,, etcKKKKK

wH

0D 0D

wH

),1830(),1800(),1760(),1460( KK

J.F. Donoghue et.al, Phys. Rev. D33, 179 (1986)

Long-distance contributions can severely enhance the mixing parameters, although it is difficult to calculate them accurately.

E. Golowich, A.A. Petrov, Phys. Lett. B427, 172 (1998)

It is estimated that long-distance dynamics can enhance the mixing parameters to be

34 1010~, yx

Contribution of higher-dimension operator in operator product expansion (OPE)

])}0()({1

Re[2

1

1

0110

020

DHxHTDdxim

DHDm

m

cw

cw

D

cw

D

})}0()({Im[1 0110 DHxHTDdxi

mc

wc

wD

The time ordered product can be expanded in local operators of increasing dimension.

The higher dimension operators are suppressed by powers of

cm/

Higher order terms in the OPE can be important, because chiral suppression can be lifted by the quark condensates, which lead to contribution proportional to ms

2, rather than ms4.

Diagram for 6-quark operator (D=9)

Diagram for 8-quark operator (D=12)

I.I. Bigi, N.G. Uraltsev, Nucl. Phys. B592 (2001)92

A. Falk et al., Phys. Rev. D65 (2002) 054034

Explicitly, the contribution of 6-quark operator with D=9 is given by

The dominant contribution to is from 6- and 8-quark operators x

For , the dominant contribution is from 8-quark operators y

Numerically, the resulting estimates are

)10(~, 3Oyx

This estimate means that the observation of mixing parameters severely larger than 10-3 would reveal the existence of new physics beyond the SM

But this statement can not be conclusive because the uncertainty in this estimation is still large.

The main uncertainty comes from the size of the relevant hadronic matrix elements. Larger observed values of x and y might indicate serious underestimation of the hadronic matrix elements.

IV Decays of physical neutral D system

How to measure the mixing parameters in experiments?

To measure them, one has to study the evolution and various decays of the neutral D meson system.

The evolution of neutral D meson state

For any neutral D state at any time t, its wave function can be expanded as linear combination of the eigenstates

2

)2

(

21

)2

(

1

2211

)( DecDectDt

imit

imi

constant

c1 and c2 can be determined by initial and normalization condition

0)2

(

1

)2

(

20)

2(

2

)2

(

1

00)2

(

200)

2(

1

2

)2

(

21

)2

(

1

)()(

)()(

)(

11222211

2211

2211

DececqDececp

DqDpecDqDpec

DecDectD

ti

miti

miti

miti

mi

ti

miti

mi

ti

miti

mi

For a state initially ,i.e., 0D 0

0|)( DtD

t

pcc

2

121

Denote such a state as )(0 tDphys

0)2

()2

(

0)2

()2

(0

)(2

)(2

1)(

1122

1122

Deep

q

DeetD

ti

miti

mi

ti

miti

mi

phys

Then

)(2

1)(

)2

()2

( 1122 ti

miti

mieetg

000 )()()( Dtgp

qDtgtDphys

000 )()()( Dtgq

pDtgtDphys

Similarly

The time-dependent decay rate of a physical neutral D meson state

The time-dependent amplitude of initial D0 physical state to any final state f

00

0

)()(

)()(

DHftgp

qDHftg

tDHftT physf

fA fA

ff Atgp

qAtg )()(

)]}sin(]Im[2)sinh(])(Re[2

)cos()()cosh(){(2

1)(

**

22

222

txAAp

qtyA

p

qA

txAp

qAtyA

p

qAetT

ffff

fffft

f

∴

20

)(])([

tTdt

ftDdf

phys

)]}sin(]Im[2)sinh(])(Re[2

)cos()()cosh(){(])([

**

22

22

0

txAAp

qtyA

p

qA

txAp

qAtyA

p

qA

Ne

dtftDd

ffff

fffff

t

phys

A normalization factor

22

,,2

12

1221

y

mmmx

00 DD

We can get the decay rate of an initial D0 state by

pq

ff AA

)]}sin(]Im[2)sinh(]Re[2

)cos()()cosh(){(])([

**

22

220

txAAq

ptyAA

q

p

txAAq

ptyAA

q

p

Ne

dtftDd

ffff

fffff

t

phys

(1) Semi-leptonic decay

At leading order of weak interaction

XlD 0 XlD 0

c

u

sd ,

u

l

0D

XlD 0 XlD 0

0lA 0lA∴

c

u u

sd ,

l

0D

With mixing present, the wrong-sign decay can occur

Let us consider the wrong-sign decays

XltDphys )(0

XltDphys )(0

)]cos()[cosh(])([

20

txtyAp

qeN

dt

XltDdl

tf

phys

)]cos()[cosh(])([

20

txtyAq

peN

dt

XltDdl

tf

phys

dt

XltDd

dt

XltDd physphys ])([])([ 00

CP

CP is violated!

llAA

CP violation in wrong-sign semileptonic decay

dtXltDddtXltDd

dtXltDddtXltDdtA

physphys

physphysSLCP /))((/))((

/))((/))(()(

00

00

4

4

1

1

pq

pq

ASLCP

We can get

The condition for mixing induced CP violation pq

c

u

sd ,

u

l

0D

c

u u

sd ,

l

0D

(2) Wrong-sign decay to hadronic final state f

For Kf

udVd

*csV

c

u

s

u

0D K

u

1)( *0 csudVVKDA

Cabibbo favored

dc

u u

us

0D

K

cdV

*usV 42*0 )( AVVKDA uscd

Doubly Cabibbo suppressed

fA

fA

Let us consider the doubly cabibbo suppressed decay

KtDphys )(0

)]}sin(]Im[2)sinh(]Re[2

)cos()()cosh(){(])([

**

22

220

txAAq

ptyAA

q

p

txAAq

ptyAA

q

p

Ne

dtftDd

ffff

fffff

t

phys

Using the formua we have derived

We can obtain

)]}sin(]Im[2)sinh(]Re[2

)cos()1()cosh()1{(

)]}sin(]Im[2)sinh(]Re[2

)cos()()cosh(){(

])([

222

**

22

22

0

txA

A

p

qty

A

A

p

q

txA

A

p

qty

A

A

p

qA

q

p

txAAq

ptyAA

q

p

txAAq

ptyAA

q

p

Ne

dtKtDd

K

K

K

K

K

K

K

KK

KKKK

KKKK

ft

phys

K

KK A

A

p

qdefine

)]}sin(]Im[2)sinh(]Re[2

)cos()1()cosh()1{(

])([

222

0

txty

txtyAq

p

Ne

dtKtDd

KK

KKK

ft

phys

)]}sin(]Im[2)sinh(]Re[2

)cos()1()cosh()1{(

])([

222

0

txty

txtyAp

q

Ne

dtKtDd

KK

KKK

ft

phys

Similarly the CP conjugated process

K

KK A

A

q

p

To simplify the above result, we can make the following definitions

'',,)1(

i

K

Ki

K

Kim er

A

Aer

A

AeA

p

q

Mixing induced CPV

In order to demonstrate the CP violation in decay, we define

)(

)(

1

1

1

1

i

M

DD

K

KK

i

D

MD

K

KK

eA

AR

A

A

q

p

eA

AR

A

A

p

q

where 2

,2

MA CP violation in mixing

DA CP violation in decay

fD 0 fD 0

fDD 00

fDD 00

CP violation in the interference between

decays with and without mixing0D f

0D

1.

2.

3.

Then the time-dependent decay rates are

})(4

)({

})(4

)))(sin()cos((1

1

1

1{

])([)(

2222

222

2

22

0

tyx

tyRRAq

p

tyx

txyRA

AR

A

AA

q

p

Ne

dtKtDdtr

DDK

DD

MD

D

MK

ft

phys

})(4

)({

])([)(

2222

0

tyx

tyRRAp

q

Ne

dtKtDdtr

DDK

ft

phys

)sin()cos(

)sin()cos(

1

1,

1

12

2

2

2

xyy

yxx

RA

ARR

A

AR D

M

DDD

D

MD

Note the sign

If CP is conserved, then

0,0,0 DM AA

Then

sincos

sincos

xyyy

yxxx

RR DD

})(4

{)()( 222

tyx

tyRRtrtr DD

The coefficients of the time-dependent terms can be fitted if the time dependent decay rate can be measured.

},,{ yxRDCP:

KtDphys )(0

KtDphys )(0for

CP: },,{ yxRD

In the early stage of 2007, BaBar and Belle found the evidence of D0-D0 mixing in the decay modes

KD0 00SKDand

respectively. Later CDF also found the evidence.

Fit type Parameter Fit results (10-3)

No CP violation DR 10.016.003.3 2x 21.030.022.0

y 1.34.47.9

CPV allowed DR 10.016.003.3 DA 155221 2

x 30.043.024.0

y 5.44.68.9

2x 29.041.020.0

y 3.41.67.9

CPV is consistent with zero

BaBar Collaboration, PRL98,211802 (2007)

Table from Belle collaboration, PRL99, 131803 (2007)

The data disfavor the no-mixing point x=0, y=0 significantly.

CP violation is also searched for in mixing and decays, but no evidence for CPV is found.

The comparison of the data and theoretical prediction is premature, because the errors of the data and the uncertainty of the theoretical prediction are both too large.

No new physics has to be invoked right now.

For a physical process producing D0 D0 such as

The decay rate of a correlated state

00'' DDee

The D0 D0 pair will be a quantum-correlated state

e e

0D

0D

The quantum number of is '' 1PCJ

∴ The C number of D0 D0 pair in this process is C

For a correlated state with C

)(2

1 0000 DDDD 00

00

ˆ

ˆ

DDC

DDC

Let us consider the decay of such a correlated state, one D decays to f1 at time t1 , the other decays to f2 at time t2

1t

2t

The correlated wave function at time t1 and t2

))()()()((2

1),( 2

01

02

01

021 tDtDtDtDtt physphysphysphys

The amplitude of such state decaying to final state is

21 ff

))()(

)()((2

1

),(

20

210

1

20

210

1

2121

tDHftDHf

tDHftDHf

ttHff

physphys

physphys

12

2121

2121

ttt

AAAAa

AAAAa

ffff

ffff

)]sin()Im(2)sinh()Re(2

)cos()()cosh()[(4

1

),(

**

2

1

22

1

2)(

2

2121

21

txaatyaa

txaatyaae

ttHff

tt

00)3770( DDee

The most promising place to produce D0D0 pair is

However, the time-dependent information can not be used here

V The time-dependent measurement of D0D0 Mixing parameters

Many interesting ideas and phenomenological studies for measuring the mixing parameters have been presented in the literature

Z.Z. Xing (1996); Z.Z. Xing, S. Zhou (2007); Y. Grossman, A. Kagan, Y. Nir (2007); N. Sinha, R. Sinha, T. Browder, N. Deshpanda, S. Pakvasa (2007);

H.B Li and M.Z. Yang PRD74 (2006) 094016; PRD75 (2007) 094015

We proposed a new method to measure the mixing parameters in the correlated time-dependent processes through

00)1( DDSee

at super-B factor

00)1( DDS

The D mesons are strongly boosted

1r

2r

Proper time interval can be precisely determined

t

Decay point 1

Decay point 2

1t

2t

12 ttt

The time-dependent evolution

);,()3( tKXl

);,()5( 21 tXlXl

);,()7( tSK

);,()9( tKK

);,()11( 21 tXlXl

);,()4( tKXl

);,()6( tSXl

);,()8( _ tKK

);,()10( __ tKK

);,()12( 21 tXlXl

as

)22(||||

);,(2222 tRtyRReAAN

tKXlR

MDDt

Kl

Fitting data, three numbers can be measured

The same thing as case (1)

(3) );,( tKXl

)4

)sincos(1( 2222

txy

txyRCe Dt

Fitting data, these two numbers can be measured

(4) );,( tsXl

)1( 222 tytyCe t

y can be measured

);,()5( tSK

))(2

11(

))(2

11()cos(

);,(

22

22222

tytyeC

tytyeRAAN

tSKR

t

tDSK

y can be measured

),sincos( xyRD

y

There are four unknown variables in these fitted numbers：

DRyx ,,,

,DR ,4

22 yx

,4

22 yx

In summary, the numbers that can be fitted are:

Therefore, the mixing parameters can be measured in the these correlated decay processes.

•In the SM CP violation is produced by the complex phase of the CKM matrix, current experimental constraint for CKM matrix element implies that CP violation in D system is very small

•However, New physics may induce new source of CP violation

•One possibility is from the new CPV phase

)/arg( pAAq

CP violation

);,();,();,();,(

)(tKXlRtKXlRtKXlRtKXlR

tACP

txyRtA DCP sin)cossin()(

One can obtain

Very small in the SM, unless there is New Physics

In experiment

KEK-B can move to Y(1S) peak)1(101.7 9 Sabout events 1 year

Super-B about 1012 Y(1S) 1 year

if the luminosity is about 1036 cm-2s-1

54 10~10))1(( DDSBr

Theoretical estimate

At super-B: 0087 10~10 DD pairs

It is possible to measure DD mixing parameters and CPV in charm by using the time-dependent information in coherent neutral DD decays

Summary

The mixing parameters can be predicted to be

310~, yx

without introducing new physics beyond the standard model.

The present uncertainty in both theoretical prediction and the exp. data are still large.

Various decays of the neutral D-meson system are studied, a new method to measure the mixing parameter by using the correlation and time-dependent information is proposed.