d 0 -d 0 mixing
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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.
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