



(Phys. Lett. B)

D

0

! K

\Sigma

ss

\Upsilon

and CP Violation at a o/ -charm Factory

Zhi-zhong Xing

1

Sektion Physik, Theoretische Physik, Universit"at M"unchen,

Theresienstrasse 37, D-80333 M"unchen, Germany

Abstract We calculate the time-dependent and time-integrated decay rates of (D

0

_

D

0

) !

(K

\Sigma

ss

\Upsilon

)(l

\Sigma

X

\Upsilon

) at the (3:77) and (4:14) resonances. The possibilities to distinguish

between the effects of D

0

\Gamma

_ D

0

mixing and doubly Cabibbo suppressed decay (DCSD)

are illustrated, and the signal of CP violation induced by the interplay of mixing and DCSD is discussed. Ratios of the decay rates of (D

0

_

D

0

) ! (K

\Sigma

ss

\Upsilon

)(K

\Sigma

ss

\Upsilon

) to that of

(D

0

_

D

0

) ! (K

+

ss

\Gamma

)(K

\Gamma

ss

+

) are also recalculated by accommodating CP violation and

final-state interactions.

1

Electronic address: Xing@hep.physik.uni-muenchen.de

1

1. Introduction

Recently some attention has been paid to the potential of searching for large D

0

\Gamma

_ D

0

mixing that is out of reach of the standard-model limitation, and to the possibility of probing significant CP violation in the charm sector [1, 2, 3, 4, 5]. The main experimental scenarios include the fixed target facilities, the o/ -charm factories and the B-meson factories [6, 7, 8, 9]. Among various decay channels of neutral D mesons, D

0

vs

_ D

0

! K

\Sigma

ss

\Upsilon

are of particular

interest, in both theory and experiments, to study the effects of D

0

\Gamma

_ D

0

mixing and doubly

Cabibbo suppressed decay (DCSD).

In the standard model, the transitions D

0

! K

\Gamma

ss

+

and

_ D

0

! K

+

ss

\Gamma

are Cabibbo favored.

In contrast, D

0

! K

+

ss

\Gamma

and

_ D

0

! K

\Gamma

ss

+

are DCSD's. If there exists detectable D

0

\Gamma

_ D

0

mixing due to new physics, then the processes D

0

!

_ D

0

! K

+

ss

\Gamma

and

_ D

0

! D

0

! K

\Gamma

ss

+

may compete with or even dominate over the respective DCSD's. Since any new physics is not likely to affect the direct decays of c quark (via the tree-level W -mediated diagrams) in a significant way [5, 10], one can use D

0

vs

_ D

0

! K

\Sigma

ss

\Upsilon

to explore the magnitude of D

0

\Gamma

_ D

0

mixing as well as the effect of CP violation induced by the interplay of decay and mixing. Of course, this idea has been extensively considered in the literature (see, e.g., refs. [1, 5]). But most of the previous studies focused on the incoherent decays of D

0

and

_ D

0

mesons, a case

applicable to the fixed target experiments (perhaps also applicable to the experiments at a B-meson factory) [6, 9].

In this work we calculate the time-dependent and time-integrated transition probabilities of coherent (D

0

_

D

0

) decays to (K

\Sigma

ss

\Upsilon

)(l

\Sigma

X

\Upsilon

), where the semileptonic final states serve to tag

the flavors of the parent D mesons, for various possible measurements at a o/ -charm factory. We illustrate how to distinguish between D

0

\Gamma

_ D

0

mixing and DCSD effects in the time

distributions of decay rates, and how to isolate the signal of CP violation arising from the interplay of these two effects. Accommodating CP violation and nonvanishing strong phase shift in D ! Kss, a recalculation of the coherent decays (D

0

_

D

0

) ! (K

\Sigma

ss

\Upsilon

)(K

\Sigma

ss

\Upsilon

) is also

given to probe D

0

\Gamma

_ D

0

mixing and to separate it from the DCSD effect. Similar analyses

can be carried out for a variety of neutral D decays to non-CP eigenstates.

2. Master formulas

At a o/ -charm factory, the coherent (D

0

_

D

0

) events can be produced through [7]

e

+

e

\Gamma

! (3:77) ! (D

0

_

D

0

)

C=\Gamma

;

e

+

e

\Gamma

! (4:14) ! fl(D

0

_

D

0

)

C=+

or ss

0

(D

0

_

D

0

)

C=\Gamma

;

2

where C represents the charge-conjugation parity. Since D

0

and

_ D

0

mix coherently until one

of them decays, one may use the semileptonic decay of one meson to tag the flavor of the other meson decaying to a flavor-nonspecific hadronic state. The time-dependent wave function for a (D

0

_

D

0

)

C

pair at rest is written as

1 p

2

h

jD

0

(k; t)i \Omega j

_ D

0

(\Gamma k; t)i + CjD

0

(\Gamma k; t)i \Omega j

_ D

0

(k; t)i

i

; (1)

where k is the three-momentum vector of D

0

and

_ D

0

mesons. The proper-time evolution of

an initially (t = 0) pure D

0

or

_ D

0

is given by

jD

0

(t)i = f

+

(t)jD

0

i + e

+i2OE

m

f

\Gamma

(t)j

_ D

0

i ;

j

_ D

0

(t)i = f

+

(t)j

_ D

0

i + e

\Gamma i2OE

m

f

\Gamma

(t)jD

0

i ;

(2)

where OE

m

is a complex parameter connecting the flavor eigenstates to the mass eigenstates

through

2

jD

L

i = e

\Gamma iOE

m

jD

0

i + e

+iOE

m

j

_ D

0

i ;

jD

H

i = e

\Gamma iOE

m

jD

0

i \Gamma e

+iOE

m

j

_ D

0

i ;

(3a)

and the evolution functions f

\Sigma

(t) read

f

\Sigma

(t) =

1

2

e

\Gamma (im+\Gamma =2)t

h

e

+(i\Delta m\Gamma \Delta \Gamma =2)t=2

\Sigma e

\Gamma (i\Delta m\Gamma \Delta \Gamma =2)t=2

i

: (3b)

In the above equation, we have defined m j (m

L

+ m

H

)=2, \Gamma j (\Gamma

L

+ \Gamma

H

)=2, \Delta m j m

H

\Gamma m

L

and \Delta \Gamma j \Gamma

L

\Gamma \Gamma

H

, where m

L(H)

and \Gamma

L(H)

are the mass and width of D

L(H)

respectively.

Now we consider the case that one D meson decays to a semileptonic state jl

+

X

\Gamma

i or

jl

\Gamma

X

+

i at (proper) time t

1

and the other to jK

+

ss

\Gamma

i or jK

\Gamma

ss

+

i at t

2

. After a lengthy

calculation, the joint decay rates for having such events are obtained as

R(l

+

; t

1

; K

+

ss

\Gamma

; t

2

)

C

/ jA

l

j

2

jA

Kss

j

2

e

\Gamma \Gamma t

+

h

(1 + j*

Kss

j

2

) cosh

i

\Delta \Gamma

2

t

C

j

\Gamma 2Re*

Kss

sinh

i

\Delta \Gamma

2

t

C

j

+ (1 \Gamma j*

Kss

j

2

) cos(\Delta mt

C

) \Gamma 2Im*

Kss

sin(\Delta mt

C

)

i

;

(4a)

R(l

\Gamma

; t

1

; K

\Gamma

ss

+

; t

2

)

C

/ jA

l

j

2

jA

Kss

j

2

e

\Gamma \Gamma t

+

h

(1 + j

~ *

Kss

j

2

) cosh

i

\Delta \Gamma

2

t

C

j

\Gamma 2Re

~ *

Kss

sinh

i

\Delta \Gamma

2

t

C

j

+ (1 \Gamma j

~ *

Kss

j

2

) cos(\Delta mt

C

) \Gamma 2Im

~ *

Kss

sin(\Delta mt

C

)

i

;

(4b)

and

R(l

\Gamma

; t

1

; K

+

ss

\Gamma

; t

2

)

C

/ jA

l

j

2

jA

Kss

j

2

e

\Gamma 4ImOE

m

e

\Gamma \Gamma t

+

h

(1 + j*

Kss

j

2

) cosh

i

\Delta \Gamma

2

t

C

j

\Gamma 2Re*

Kss

sinh

i

\Delta \Gamma

2

t

C

j

\Gamma (1 \Gamma j*

Kss

j

2

) cos(\Delta mt

C

) + 2Im*

Kss

sin(\Delta mt

C

)

i

;

(5a)

R(l

+

; t

1

; K

\Gamma

ss

+

; t

2

)

C

/ jA

l

j

2

jA

Kss

j

2

e

+4ImOE

m

e

\Gamma \Gamma t

+

h

(1 + j

~ *

Kss

j

2

) cosh

i

\Delta \Gamma

2

t

C

j

\Gamma 2Re

~ *

Kss

sinh

i

\Delta \Gamma

2

t

C

j

\Gamma (1 \Gamma j

~ *

Kss

j

2

) cos(\Delta mt

C

) + 2Im

~ *

Kss

sin(\Delta mt

C

)

i

:

(5b)

2

Here CP T symmetry in the D

0

\Gamma

_ D

0

mixing matrix has been assumed.

3

In deriving these formulas, we have used the \Delta Q = \Delta C rule and CP T invariance as well as the reliable assumption that there is no direct CP violation in the decay amplitudes of D

0

! K

\Sigma

ss

\Upsilon

. The relevant quantities appearing in eqs. (4) and (5) are defined as follows:

t

C

j t

2

+ Ct

1

, A

l

j hl

+

X

\Gamma

jHjD

0

i, A

Kss

j hK

\Gamma

ss

+

jHjD

0

i, *

Kss

j exp(\Gamma 2iOE

m

) ,

Kss

and

~ *

Kss

j exp(+2iOE

m

)

~ ,

Kss

, where

,

Kss

j

hK

+

ss

\Gamma

jHjD

0

i

hK

+

ss

\Gamma

jHj

_ D

0

i

;

~ ,

Kss

j

hK

\Gamma

ss

+

jHj

_ D

0

i

hK

\Gamma

ss

+

jHjD

0

i

: (6)

Note that j

~ ,

Kss

j = j,

Kss

j holds in the absence of direct CP violation. But in general

~ ,

Kss

= ,

\Lambda

Kss

is not true due to the existence of final-state interactions [5]. Moreover, one should keep in

mind that j,

Kss

j is doubly Cabibbo suppressed and its magnitude is of the order 10

\Gamma 2

[1].

It is known that ImOE

m

6= 0 implies observable CP violation in D

0

\Gamma

_ D

0

mixing. This kind

of CP -violating signal can manefest itself in the like-sign dilepton events of (D

0

_

D

0

)

C

pairs:

N

\Gamma \Gamma

C

j R(l

\Gamma

; t

1

; l

\Gamma

; t

2

)

C

/ e

\Gamma 4ImOE

m

e

\Gamma \Gamma t

+

h

cosh

i

\Delta \Gamma

2

t

C

j

\Gamma cos(\Delta mt

C

)

i

;

N

++

C

j R(l

+

; t

1

; l

+

; t

2

)

C

/ e

+4ImOE

m

e

\Gamma \Gamma t

+

h

cosh

i

\Delta \Gamma

2

t

C

j

\Gamma cos(\Delta mt

C

)

i

:

(7)

If ImOE

m

is of the level O(* 10

\Gamma 3

), it should be first observed from the CP -violating asymmetry

N

++

C

\Gamma N

\Gamma \Gamma

C

N

++

C

+ N

\Gamma \Gamma

C

=

e

+4ImOE

m

\Gamma e

\Gamma 4ImOE

m

e

+4ImOE

m

+ e

\Gamma 4ImOE

m

ss 4ImOE

m

: (8)

Note that this asymmetry is not only independent of the time distributions of N

\Sigma \Sigma

C

but

also independent of the charge-conjugation parity C, thus it can be measured by using the time-integrated dilepton events at either (3:77) or (4:14) resonance.

In the following discussions about the decay modes D

0

! K

\Sigma

ss

\Upsilon

and their CP -conjugate

processes, we shall neglect the contribution from ImOE. We shall in turn use the assumption j\Delta \Gamma j !! j\Delta mj (i.e., neglecting the mixing effect induced by \Delta \Gamma ), which is very likely to be a good approximation if j\Delta mj is close to its current experimental bound [5]. As a consequence, j

~ *

Kss

j = j*

Kss

j holds.

3. Time dependence of the decay rates

By use of the approximations mentioned above, the formulas in eqs. (4) and (5) can be simplified. Here we assume that a reconstruction of the decay-time differences t

\Gamma

between

neutral D decays to l

\Sigma

X

\Upsilon

and K

\Sigma

ss

\Upsilon

is possible in experiments. Usually it is difficult to detect

the t

+

distribution of joint decay rates in either linacs or storage rings, since the creation point

of the (3:77) or (4:14) resonance cannot be well resolved [11]. Hence we prefer to integrate R(l

\Sigma

; t

1

; K

\Sigma

ss

\Upsilon

; t

2

)

C

over t

+

in order to obtain the t

\Gamma

distributions. For simplicity, we define

4

a dimensionless parameter T j \Gamma t

\Gamma

and choose T ? 0 by convention. The relevant results

are given as follows:

R(l

+

; K

+

ss

\Gamma

; T )

\Gamma

/ e

\Gamma T

(4 \Gamma x

2

T

2

\Gamma 4xT Im*

Kss

) ;

R(l

\Gamma

; K

\Gamma

ss

+

; T )

\Gamma

/ e

\Gamma T

(4 \Gamma x

2

T

2

\Gamma 4xT Im

~ *

Kss

) ;

R(l

\Gamma

; K

+

ss

\Gamma

; T )

\Gamma

/ e

\Gamma T

(4j*

Kss

j

2

+ x

2

T

2

+ 4xT Im*

Kss

) ;

R(l

+

; K

\Gamma

ss

+

; T )

\Gamma

/ e

\Gamma T

(4j

~ *

Kss

j

2

+ x

2

T

2

+ 4xT Im

~ *

Kss

) ;

(9)

and

R(l

+

; K

+

ss

\Gamma

; T )

+

/ e

\Gamma T

[4 \Gamma x

2

(2 + 2T + T

2

) \Gamma 4x(1 + T )Im*

Kss

] ;

R(l

\Gamma

; K

\Gamma

ss

+

; T )

+

/ e

\Gamma T

[4 \Gamma x

2

(2 + 2T + T

2

) \Gamma 4x(1 + T )Im

~ *

Kss

] ;

R(l

\Gamma

; K

+

ss

\Gamma

; T )

+

/ e

\Gamma T

[4j*

Kss

j

2

+ x

2

(2 + 2T + T

2

) + 4x(1 + T )Im*

Kss

] ;

R(l

+

; K

\Gamma

ss

+

; T )

+

/ e

\Gamma T

[4j

~ *

Kss

j

2

+ x

2

(2 + 2T + T

2

) + 4x(1 + T )Im

~ *

Kss

] ;

(10)

where x j \Delta m=\Gamma is a D

0

\Gamma

_ D

0

mixing parameter. In obtaining eqs. (9) and (10), we have

assumed x to be at the detectable level [12] (i.e., jxj , 10

\Gamma 2

) and made approximations up to

the accuracy of O(x

2

), O(j*

Kss

j

2

) or O(xj*

Kss

j).

One can observe that the D

0

\Gamma

_ D

0

mixing (x) and DCSD (*

Kss

or

~ *

Kss

) terms play

insignificant roles in the decay rates R(l

+

; K

+

ss

\Gamma

; T )

C

and R(l

\Gamma

; K

\Gamma

ss

+

; T )

C

. In contrast,

R(l

\Gamma

; K

+

ss

\Gamma

; T )

C

and R(l

+

; K

\Gamma

ss

+

; T )

C

are remarkably suppressed due to the smallness of

x and j*

Kss

j. Let us parametrize ,

Kss

and

~ ,

Kss

as follows: ,

Kss

= j,

Kss

je

i(ffi

Kss

+OE

t

)

and

~ ,

Kss

=

j,

Kss

je

i(ffi

Kss

\Gamma OE

t

)

, where ffi

Kss

and OE

t

stand for the strong phase shift and the weak transition

phase respectively. Then Im*

Kss

and Im

~ *

Kss

are given as

Im*

Kss

= j,

Kss

j sin(ffi

Kss

+ OE

t

\Gamma 2OE

m

) ;

Im

~ *

Kss

= j,

Kss

j sin(ffi

Kss

\Gamma OE

t

+ 2OE

m

) :

(11)

In the standard model, we have OE

t

= arg[(V

cd

V

\Lambda

us

)=(V

ud

V

\Lambda

cs

)] ss A

2

*

4

j ^ 10

\Gamma 3

as well as

OE

m

= arg[(V

us

V

\Lambda

cs

)=(V

cs

V

\Lambda

us

)]=2 ss 0, where A; * and j are the Wolfenstein parameters. Since

any new physics is unlikely to significantly affect the direct decays of c quark, OE

t

ss 0 is always

a good approximation and will be used later on. New physics might introduce additional non- trivial phases into OE

m

[10, 13], leading to observable CP -violating effects through the interplay

of decay and mixing.

To isolate the mixing and DCSD effects, the following two types of measurables can be analyzed in experiments:

(a) The combined decay rate

\Omega

+\Gamma

C

(T ) j R(l

\Gamma

; K

+

ss

\Gamma

; T )

C

+ R(l

+

; K

\Gamma

ss

+

; T )

C

:

5

Explicitly, we get

\Omega

+\Gamma

\Gamma

(T ) / 2 e

\Gamma T

[4j,

Kss

j

2

+ x

2

T

2

+ 4xj,

Kss

jT sin ffi

Kss

cos(2OE

m

)] ;

\Omega

+\Gamma

+

(T ) / 2 e

\Gamma T

[4j,

Kss

j

2

+ x

2

(2 + 2T + T

2

) + 4xj,

Kss

j(1 + T ) sin ffi

Kss

cos(2OE

m

)] :

(12)

Note that the strong phase shift ffi

Kss

vanishes only in the limit of SU(3) symmetry [14]. To

fit the CLEO II result for D

0

! K

\Sigma

ss

\Upsilon

, which gives j,

Kss

j

2

= (0:77 \Sigma 0:25 \Sigma 0:25)% [15], one

finds ffi

Kss

, 5

0

\Gamma 13

0

from a few phenomenological models [16]. For illustration, we show

the time dependence of \Omega

+\Gamma

C

(T ) in Fig. 1 by taking j,

Kss

j = 0:08, ffi

Kss

= 10

0

, OE

m

= 30

0

and

x = 0:06 (the upper bound of jxj is 0.086 [12]). One can observe that \Omega

+\Gamma

\Gamma

(T ) and \Omega

+\Gamma

+

(T )

are most sensitive to the presence of DCSD or D

0

\Gamma

_ D

0

mixing in the regions T , 1 \Gamma 4

and T , 0 \Gamma 4, respectively. For small values of T , the signal of mixing in \Omega

+\Gamma

+

(T ) is more

significant than that in \Omega

+\Gamma

\Gamma

(T ). Thus it is in principle favourable to study D

0

\Gamma

_ D

0

mixing

by use of (D

0

_

D

0

)

C=+

events at the (4:14) resonance.

(b) The CP -violating asymmetry

A

+\Gamma

C

(T ) j

R(l

+

; K

+

ss

\Gamma

; T )

C

\Gamma R(l

\Gamma

; K

\Gamma

ss

+

; T )

C

R(l

+

; K

+

ss

\Gamma

; T )

C

+ R(l

\Gamma

; K

\Gamma

ss

+

; T )

C

:

This signal arises from the interplay of DCSD and mixing. With the help of eqs. (9) and (10), we obtain

A

+\Gamma

\Gamma

(T ) ss xj,

Kss

jT cos ffi

Kss

sin(2OE

m

) ;

A

+\Gamma

+

(T ) ss xj,

Kss

j(1 + T ) cos ffi

Kss

sin(2OE

m

) :

(13)

It is clear that the above asymmetries are considerably suppressed due to the smallness of x and j,

Kss

j. In addition, nonvanishing OE

m

is a necessary condition to have nonzero A

+\Gamma

C

(T ).

We illustrate the changes of A

+\Gamma

C

(T ) with T in Fig. 2, where the inputs are the same as in

case (a). Here it should be noted that large data samples for (l

+

; K

+

ss

\Gamma

) and (l

\Gamma

; K

\Gamma

ss

+

)

events are available at a o/ -charm factory, since such joint decay rates are not suppressed by small mixing and DCSD effects (see eqs. (9) and (10)). Hence it is quite likely to measure A

+\Gamma

C

(T ) in the near future, if the magnitudes of x and OE

m

are large enough (e.g., x * 0:05

and sin(2OE

m

) * 0:5).

A natural question to be asked is if the decay-time differences T (or t

\Gamma

) between the

semileptonic and nonleptonic D decays can really be measured on the (3:77) and (4:14) resonances. For symmetric e

+

e

\Gamma

collisions at the \Upsilon (4S) resonance, the produced B

0

d

_ B

0

d

pair is

almost at rest so that their mean decay length is insufficient for identifying the decay vertices or measuring the decay-time difference [11]. Hence designing an asymmetric B factory is necessary to study the time distribution of joint (B

0

d

_ B

0

d

) decays and CP violation. For the

similar kinetic reasons, a symmetric o/ -charm factory might also be unable to resolve the decay

6

vertices of (D

0

_

D

0

) events, and this would make the time-dependent measurements discussed

above impossible in practice. Of course, an instructive analysis of this problem is desirable to give definite conclusions for the o/ -charm factory designers. From our point of view it is suggestive to have an asymmetric o/ -charm factory running at the (3:77) and (4:14) resonances, which can allow various possible measurements of D

0

\Gamma

_ D mixing, DCSD and

CP -violating effects in coherent (D

0

_

D

0

)

C

decays.

4. Time integration of the decay rates

In this section we calculate the time-integrated probabilities of the joint decays (D

0

_

D

0

)

C

!

(l

\Sigma

X

\Upsilon

)(K

\Sigma

ss

\Upsilon

). Without loss of any generality, we start from eqs. (4) and (5), where no

special approximation has been made. Integrating R(l

\Sigma

; t

1

; K

\Sigma

ss

\Upsilon

; t

2

)

C

over t

1

2 [0; +1) and

t

2

2 [0; +1), one obtains:

R(l

+

; K

+

ss

\Gamma

)

C

/ jA

l

j

2

jA

Kss

j

2

"

1 + Cy

2

(1 \Gamma y

2

)

2

(1 + j*

Kss

j

2

) \Gamma

2(1 + C)y

(1 \Gamma y

2

)

2

Re*

Kss

+

1 \Gamma Cx

2

(1 + x

2

)

2

(1 \Gamma j*

Kss

j

2

) \Gamma

2(1 + C)x

(1 + x

2

)

2

Im*

Kss

#

;

(14a)

R(l

\Gamma

; K

\Gamma

ss

+

)

C

/ jA

l

j

2

jA

Kss

j

2

"

1 + Cy

2

(1 \Gamma y

2

)

2

(1 + j

~ *

Kss

j

2

) \Gamma

2(1 + C)y

(1 \Gamma y

2

)

2

Re

~ *

Kss

+

1 \Gamma Cx

2

(1 + x

2

)

2

(1 \Gamma j

~ *

Kss

j

2

) \Gamma

2(1 + C)x

(1 + x

2

)

2

Im

~ *

Kss

#

;

(14b)

and

R(l

\Gamma

; K

+

ss

\Gamma

)

C

/ jA

l

j

2

jA

Kss

j

2

e

\Gamma 4ImOE

m

"

1 + Cy

2

(1 \Gamma y

2

)

2

(1 + j*

Kss

j

2

) \Gamma

2(1 + C)y

(1 \Gamma y

2

)

2

Re*

Kss

\Gamma

1 \Gamma Cx

2

(1 + x

2

)

2

(1 \Gamma j*

Kss

j

2

) +

2(1 + C)x

(1 + x

2

)

2

Im*

Kss

#

;

(15a)

R(l

+

; K

\Gamma

ss

+

)

C

/ jA

l

j

2

jA

Kss

j

2

e

+4ImOE

m

"

1 + Cy

2

(1 \Gamma y

2

)

2

(1 + j

~ *

Kss

j

2

) \Gamma

2(1 + C)y

(1 \Gamma y

2

)

2

Re

~ *

Kss

\Gamma

1 \Gamma Cx

2

(1 + x

2

)

2

(1 \Gamma j

~ *

Kss

j

2

) +

2(1 + C)x

(1 + x

2

)

2

Im

~ *

Kss

#

;

(15b)

where y j \Delta \Gamma =(2\Gamma ) is another D

0

\Gamma

_ D

0

mixing parameter. ?From the above formulas one

can observe that for C = \Gamma case the interference terms Re*

Kss

, Im*

Kss

, Re

~ *

Kss

and Im

~ *

Kss

disappear in the joint decay rates. Indeed this is a generic feature of coherent (D

0

_

D

0

)

C=\Gamma

decays, independent of the final products to be semileptonic or nonleptonic states [11].

In the assumption of j\Delta \Gamma j !! j\Delta mj, one equivalently has jyj !! jxj. Subsequently we neglect the contributions from ImOE and y as well as the O(^ x

3

) terms, in order to simplify

7

eqs. (14) and (15). The relevant time-integrated decay rates turn out to be

R(l

+

; K

+

ss

\Gamma

)

\Gamma

= R(l

\Gamma

; K

\Gamma

ss

+

)

\Gamma

/ jA

l

j

2

jA

Kss

j

2

h

2 \Gamma x

2

i

1 \Gamma j*

Kss

j

2

ji

;

R(l

\Gamma

; K

+

ss

\Gamma

)

\Gamma

= R(l

+

; K

\Gamma

ss

+

)

\Gamma

/ jA

l

j

2

jA

Kss

j

2

h

2j*

Kss

j

2

+ x

2

i

1 \Gamma j*

Kss

j

2

ji

;

(16)

and

R(l

+

; K

+

ss

\Gamma

)

+

/ jA

l

j

2

jA

Kss

j

2

h

2 \Gamma 3x

2

i

1 \Gamma j*

Kss

j

2

j

\Gamma 4xIm*

Kss

i

;

R(l

\Gamma

; K

\Gamma

ss

+

)

+

/ jA

l

j

2

jA

Kss

j

2

h

2 \Gamma 3x

2

i

1 \Gamma j

~ *

Kss

j

2

j

\Gamma 4xIm

~ *

Kss

i

;

R(l

\Gamma

; K

+

ss

\Gamma

)

+

/ jA

l

j

2

jA

Kss

j

2

h

2j*

Kss

j

2

+ 3x

2

i

1 \Gamma j*

Kss

j

2

j

+ 4xIm*

Kss

i

;

R(l

+

; K

\Gamma

ss

+

)

+

/ jA

l

j

2

jA

Kss

j

2

h

2j

~ *

Kss

j

2

+ 3x

2

i

1 \Gamma j

~ *

Kss

j

2

j

+ 4xIm

~ *

Kss

i

:

(17)

Here the x

2

j*

Kss

j

2

and x

2

j

~ *

Kss

j

2

terms are further negligible. These formulas can be compared

with those given in eqs. (9) and (10).

In practical experiments, the combined decay rates

\Omega

+\Gamma

C

j R(l

\Gamma

; K

+

ss

\Gamma

)

C

+ R(l

+

; K

\Gamma

ss

+

)

C

should be observable. Similar to eq. (12), we find

\Omega

+\Gamma

\Gamma

/ 2jA

l

j

2

jA

Kss

j

2

h

2j,

Kss

j

2

+ x

2

i

1 \Gamma j,

Kss

j

2

ji

\Omega

+\Gamma

+

/ 2jA

l

j

2

jA

Kss

j

2

h

2j,

Kss

j

2

+ 3x

2

i

1 \Gamma j,

Kss

j

2

j

+ 4xj,

Kss

j sin ffi

Kss

cos(2OE

m

)

i

:

(18)

Clearly the effects of D

0

\Gamma

_ D

0

mixing and DCSD cannot be distinguished from each other, if the

sizes of jxj and j,

Kss

j are comparable. In the case of jxj !! j,

Kss

j, the relation \Omega

+\Gamma

\Gamma

= \Omega

+\Gamma

+

/

j,

Kss

j

2

holds, as expected in the standard model. Here the time-integrated CP asymmetries

can be defined as

A

+\Gamma

C

j

R(l

+

; K

+

ss

\Gamma

)

C

\Gamma R(l

\Gamma

; K

\Gamma

ss

+

)

C

R(l

+

; K

+

ss

\Gamma

)

C

+ R(l

\Gamma

; K

\Gamma

ss

+

)

C

:

Explicitly, we obtain

A

+\Gamma

\Gamma

ss 0 ;

A

+\Gamma

+

ss 2xj,

Kss

j cos ffi

Kss

sin(2OE

m

) :

(19)

This result is quite similar to that obtained for B

0

d

vs

_ B

0

d

! D

\Sigma

ss

\Upsilon

at the \Upsilon (4S) [11].

Now we roughly estimate the magnitudes of \Omega

+\Gamma

\Gamma

, \Omega

+\Gamma

+

and A

+\Gamma

+

, to give one a feeling of

ballpark numbers to be expected. Assuming the semileptonic decay mode serving for flavor tagging to be D

0

! K

\Gamma

e

+

*

e

, we have its branching ratio Br(D

0

! K

\Gamma

e

+

*

e

) ss 3:8% [12].

In addition, the current data give Br(D

0

! K

\Gamma

ss

+

) ss 4:01% [12]. Then R(l

+

; K

+

ss

\Gamma

)

\Sigma

and R(l

\Gamma

; K

\Gamma

ss

+

)

\Sigma

are at the level 10

\Gamma 3

or so, while \Omega

+\Gamma

\Sigma

may be of the order 10

\Gamma 5

if we

input x , 0:06. Within the experimental capabilities of a o/ -charm factory, it is possible to measure \Omega

+\Gamma

\Sigma

as well as \Omega

+\Gamma

\Sigma

(T ) to an acceptable degree of accuracy with about 10

7

(D

0

_

D

0

)

8

events. Furthermore, the upper bound of the CP asymmetry A

+\Gamma

+

can be obtained by use

of the experimental results jxj ! 0:086 and j,

Kss

j ss 0:088 [12, 15]. Taking cos ffi

Kss

= 1

and sin(2OE

m

) = \Sigma 1, we get jA

+\Gamma

+

j ! 0:015. Similar constraints on the time-dependent CP

asymmetries in eq. (13) are then obtainable through the relations A

+\Gamma

\Gamma

(T ) = 0:5A

+\Gamma

+

T and

A

+\Gamma

+

(T ) = 0:5A

+\Gamma

+

(1 + T ). In the assumption of perfect detectors or 100% tagging efficiencies,

one needs about 10

8

(D

0

_

D

0

) events to uncover jA

+\Gamma

+

j , 0:01 at the level of three standard

deviations or to measure jA

+\Gamma

+

j , 0:005 at the level of one standard deviation. Accumulation

of so many events is of course a serious challenge to all types of facilities for charm physics, but it should be achieved in the second-round experiments of a o/ -charm factory.

5. Further discussions

Taking into account the D

0

\Gamma

_ D

0

mixing and DCSD effects, we have calculated the time-

dependent and time-integrated decay rates for neutral D decays to K

\Sigma

ss

\Upsilon

at a o/ -charm

factory. The similarities and differences between the decay modes from (D

0

_

D

0

)

C=+

and

(D

0

_

D

0

)

C=\Gamma

events are illustrated. We have also taken a look at the CP asymmetry of

D

0

vs

_ D

0

! K

\Sigma

ss

\Upsilon

, which is induced by the interplay of D

0

\Gamma

_ D

0

mixing and DCSD. A

similar discussion can be given for other neutral D decays to two-body non-CP eigenstates with DCSD effects included, such as D

0

! K

\Sigma

ae

\Upsilon

; K

\Lambda \Sigma

ss

\Upsilon

and their CP -conjugate processes.

These channels may occur through the same weak interactions, but their final-state strong interactions should be different from one another (e.g., ffi

Kss

6= ffi

Kae

). If the SU(3) breaking

effects in D

0

! (K

\Sigma

; K

\Lambda \Sigma

) + (ss

\Upsilon

; ae

\Upsilon

; a

\Upsilon

1

; etc) are not so significant that all the strong phase

shifts lie in the same quadrant as ffi

Kss

, then a sum over these modes is possible to increase the

number of decay events in statistics, with few dilution effect on the signal of CP violation.

It has been pointed out that the coherent decays (D

0

_

D

0

)

C

! (K

\Sigma

ss

\Upsilon

)(K

\Sigma

ss

\Upsilon

) can be used

to search for D

0

\Gamma

_ D

0

mixing and to separate it from the DCSD effect [1, 17]. The relevant

measurables may be

r

+\Gamma

C

j

R(K

+

ss

\Gamma

; K

+

ss

\Gamma

)

C

R(K

\Gamma

ss

+

; K

+

ss

\Gamma

)

C

or r

\Gamma +

C

j

R(K

\Gamma

ss

+

; K

\Gamma

ss

+

)

C

R(K

\Gamma

ss

+

; K

+

ss

\Gamma

)

C

:

Since in previous calculations the effects of CP violation and nonvanishing ffi

Kss

on r

\Sigma \Upsilon

C

were

neglected, it is worth having a recalculation of these observables without special approxima- tions. Explicitly, we find

3

:

r

+\Gamma

\Gamma

ss r

\Gamma +

\Gamma

ss

1

2

i

x

2

+ y

2

j

; (20)

and

r

+\Gamma

+

ss

3

2

(x

2

+ y

2

) + 4(j*

Kss

j

2

+ xIm*

Kss

\Gamma yRe*

Kss

) ;

r

\Gamma +

+

ss

3

2

(x

2

+ y

2

) + 4(j

~ *

Kss

j

2

+ xIm

~ *

Kss

\Gamma yRe

~ *

Kss

) :

(21)

3

Here it is not necessary to assume jxj ?? jyj, but ImOE

m

= 0 has been used.

9

One can see that r

+\Gamma

+

6= r

\Gamma +

+

in general, and their difference implies the presence of CP

violation:

r

\Gamma +

+

\Gamma r

+\Gamma

+

ss 8j,

Kss

j sin(2OE

m

) (x cos ffi

Kss

+ y sin ffi

Kss

) : (22)

In the approximation of y = 0, we get r

\Gamma +

+

\Gamma r

+\Gamma

+

ss 4A

+\Gamma

+

. Of course, this CP asymmetry is

very interesting and should be searched for in experiments.

Certainly there are other possibilities to explore CP violation in neutral D decays [1, 2]. In particular, direct CP violation in the decay amplitude is expected to give rise to observable effects in some decay modes to CP eigenstates (e.g., D

0

vs

_ D ! K

+

K

\Gamma

; ss

+

ss

\Gamma

and K

S

ae)

[3, 18]. Following the same approaches outlined in this work, a detailed analysis of the time- dependent and time-integrated decay rates for (D

0

_

D

0

)

C

! (K

+

K

\Gamma

)(l

\Sigma

X

\Upsilon

) etc is worth while.

In practical experiments, however, a symmetric o/ -charm factory seems difficult to implement the time-dependent measurements. Hence one is invited to speculate an asymmetric machine and its physics potential [19].

Our conclusion is that coherent (D

0

_

D

0

) decays to (K

\Sigma

ss

\Upsilon

)(l

\Sigma

X

\Upsilon

) and (K

\Sigma

ss

\Upsilon

)(K

\Sigma

ss

\Upsilon

)

contain clear signals of D

0

\Gamma

_ D

0

mixing, DCSD and CP violation. They can be measured

through either time-dependent or time-integrated way (or both) at a o/ -charm factory running at the (3:77) or (4:14) resonance. It is also interesting to study a variety of decay modes similar to D

0

! K

\Sigma

ss

\Upsilon

at the same experimental scenario.

I would like to thank Harald Fritzsch for his warm hospitality and constant encouragement. This work was stimulated from useful discussions with Dongsheng Du, Daniel M. Kaplan and Tiehui (Ted) Liu. I am greatly indebted to Ted for his reading the manuscript and giving many constructive comments on it. Finally I acknowledge the Alexander von Humboldt Foundation of Germany for its financial support.

References

[1] I.I. Bigi and A.I. Sanda, Phys. Lett. B171 (1986) 320; I.I. Bigi, in the Proceedings of

the Tau-Charm Factory Workshop, SLAC, May 1989 (edited by L.V. Beers), p. 169; I.I. Bigi, F. Gabbiani, and A. Masiero, Z. Phys. C48 (1990) 633.

[2] D. Du, Phys. Rev. D34 (1986) 3428. [3] F. Buccella et al., Phys. Lett. B302 (1993) 319; A.L. Yaouanc, L. Oliver, and J.C. Raynal,

Phys. Lett. B292 (1992) 353. I.I. Bigi and H. Yamamoto, Phys. Lett. B349 (1995) 363.

10

[4] Z.Z. Xing, Phys. Lett. B353 (1995) 313. [5] G. Blaylock, A. Seiden, and Y. Nir, Phys. Lett. B355 (1995) 555; T.E. Browder and S.

Pakvasa, preprint UH 511-828-95 (1995).

[6] D.M. Kaplan, in the Proceedings of the Charm 2000 Workshop, Fermilab, June 1994

(edited by D.M. Kaplan and S. Kwan), p. 35; preprint IIT-HEP-95/3 (presented at the Tau-Charm Factory Workshop, Argonne, June 1995).

[7] J.R. Fry and T. Ruf, in the Proceedings of the Third Workshop on the Tau-Charm

Factory, Marbella, June 1993 (edited by J. Kirkby), p. 387.

[8] J.L. Hewett, in the Proceedings of the Tau-Charm Factory in the Era of B-Factories and

CESR, SLAC, August 1994 (edited by L.V. Beers and M.L. Perl), p. 5.

[9] T. Liu, in the Proceedings of the Charm 2000 Workshop, Fermilab, June 1994 (edited by

D.M. Kaplan and S. Kwan); preprint Princeton/HEP/95-6 (presented at the Tau-Charm Factory Workshop, Argonne, June 1995).

[10] Z.Z. Xing, Phys. Lett. B371 (1996) 310. [11] Z.Z. Xing, Phys. Rev. D53 (1996) 204; and references therein. [12] Particle Data Group, L. Montanet et al., Phys. Rev. D50 (1994) 1173. [13] Z.Z. Xing, "On Testing Unitarity of the Quark Mixing Matrix", preprint LMU-13/95

(to appear in the Proceedings of the Conference on Production and Decay of Hyperons, Charm and Beauty Hadrons, Strasbourg, September 1995).

[14] L. Wolfenstein, Phys. Rev. Lett. 75 (1995) 2460. [15] CLEO Collaboration, D. Cinabro et al., Phys. Rev. Lett. 72 (1994) 406. [16] L.L. Chau and H.Y. Cheng, Phys. Lett. B333 (1994) 514; F. Buccella et al., Phys. Rev.

D51 (1995) 3478; T. Kaeding, preprint LBL-37224 (to appear in Phys. Lett. B).

[17] H. Yamamoto, Ph.D. Thesis, preprint CALT-68-1318 (1985). [18] CLEO Collaboration, J. Bartelt et al., Phys. Rev. D52 (1995) 4860. [19] T. Liu, private communications.

11

0 1 2 3 4 5 6

0 2 4 6 8 10 \Omega

+\Gamma

\Gamma

(T )

T a b

c

(x, j,

Kss

j)

a: (0.06, 0.08) b: (0.00, 0.08) c: (0.06, 0.00)

\Omega

+\Gamma

+

(T )

T a b

c

(x, j,

Kss

j)

a: (0.06, 0.08) b: (0.00, 0.08) c: (0.06, 0.00)

0 1 2 3 4 5 6 7

0 2 4 6 8 10 Figure 1: Illustrative plots of the time dependence of \Omega

+\Gamma

\Gamma

(T ) and \Omega

+\Gamma

+

(T ) (in common but

arbitrary units), where ffi

Kss

= 10

0

and OE

m

= 30

0

are taken.

12

.00 .01 .02 .03 .04 .05

0 2 4 6 8 10 A

+\Gamma

C

(T )

T C = +

C = \Gamma

Figure 2: Illustrative plot of the time dependence of A

+\Gamma

C

(T ), where x = 0:06, j,

Kss

j = 0:08,

ffi

Kss

= 10

0

and OE

m

= 30

0

are taken.

13

