



(Phys. Lett. B)

A Time-independent Way to Probe D

0

\Gamma

_ D

0

Mixing

at o/ -charm Factories

Zhi-zhong Xing

1

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

Theresienstrasse 37, D-80333 M"unchen, Germany

and Department of Physics, Faculty of Science, Nagoya University,

Chikusa-ku, Nagoya 464-01, Japan

2

Abstract D

0

\Gamma

_ D

0

mixing leads to the mass and width differences in the mass eigenstates of D

0

and

_ D

0

mesons (measured by parameters x

D

and y

D

respectively), but their magnitudes

cannot be reliably predicted by the standard model. We show that it is possible to separately determine x

D

and y

D

through time-integrated measurements of the dilepton

events of coherent D

0

_

D

0

decays on the (4:14) resonance at a o/-charm factory.

1

Electronic address: xing@eken.phys.nagoya-u.ac.jp

2

Mailing address

1

It is known in charm physics that mixing between D

0

and

_ D

0

mesons can arise naturally,

since both of them couple to a subset of the virtual and real intermediate states. The rate of D

0

\Gamma

_ D

0

mixing is commonly measured by two well-defined parameters x

D

and y

D

, which correspond

to the mass and width differences in the mass eigenstates of D

0

and

_ D

0

. In the standard model

the short-distance (box diagram) contribution to D

0

\Gamma

_ D

0

mixing is expected to be negligibly

small [1], but different approaches to the long-distance effects have given dramatically different estimates for the magnitudes of x

D

and y

D

[2, 3, 4, 5].

The latest E691 data of Fermilab fixed target experiments give an upper bound on D

0

\Gamma

_ D

0

mixing: r

D

ss (x

2

D

+ y

2

D

)=2 ! 0:37% [6]. If x

D

and y

D

are well below 10

\Gamma 2

as predicted by

the dispersive approach of ref. [2] or the heavy quark effective theory of ref. [4], then the observation of r

D

at the level of 10

\Gamma 4

or so will imply the presence of new physics [7, 8, 9, 10].

Today much more theoretical effort is needed to make sure of the order of x

D

and y

D

. On

the experimental side, searches for D

0

\Gamma

_ D

0

mixing to a high degree of accuracy (e.g., r

D

, 10

\Gamma 4

to 10

\Gamma 5

) are expected to be available at future fixed target facilities, B-meson factories and

o/ -charm factories [11, 12, 13].

In this work we investigate how the subtlety of D

0

\Gamma

_ D

0

mixing can be probed at a o/ -charm

factory. We show that it is possible to separately determine the mixing parameters x

D

and y

D

by use of the time-integrated dilepton events of coherent D

0

_

D

0

decays at the (4:14) resonance.

The importance of such measurements is that knowledge of the relative magnitude of x

D

and y

D

can definitely clarify the ambiguities of current theoretical estimates and shed light on possible

sources of new physics in D

0

\Gamma

_ D

0

mixing.

In the presence of CP violation, the CP eigenstates jD

1

i j (jD

0

i + j

_ D

0

i)=

p

2 and jD

2

i j

(jD

0

i \Gamma j

_ D

0

i)=

p

2 are related to the mass eigenstates jD

L

i and jD

H

i by

jD

L

i = cosh(iOE)jD

1

i \Gamma sinh(iOE)jD

2

i ;

jD

H

i = cosh(iOE)jD

2

i \Gamma sinh(iOE)jD

1

i ;

(1)

where the subscripts "L" and "H" stand for Light and Heavy respectively, and OE is a complex phase. The proper-time evolution of an initially (t = 0) pure D

0

or

_ D

0

is given as

jD

0

(t)i = f

0

(t)

h

f

+

(t)jD

0

i + exp(+i2OE)f

\Gamma

(t)j

_ D

0

i

i

;

j

_ D

0

(t)i = f

0

(t)

h

f

+

(t)j

_ D

0

i + exp(\Gamma i2OE)f

\Gamma

(t)jD

0

i

i

;

(2)

2

in which the evolution functions read f

0

(t) = exp[\Gamma (im+\Gamma =2)t], f

+

(t) = cosh[(i\Delta m\Gamma \Delta \Gamma =2)t=2]

and f

\Gamma

(t) = sinh[(i\Delta m \Gamma \Delta \Gamma =2)t=2]. Here we have used the common notations 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

. Furthermore, we define

x

D

j \Delta m=\Gamma and y

D

j \Delta \Gamma =(2\Gamma ) as two characteristic parameters of D

0

\Gamma

_ D

0

mixing.

For fixed target experiments or e

+

e

\Gamma

collisions at the \Upsilon (4S) resonance, the D

0

and

_ D

0

mesons can be produced incoherently. Knowledge of D

0

\Gamma

_ D

0

mixing is expected to come from

ratios of the wrong-sign to right-sign events of semileptonic D decays:

r j

\Gamma (D

0

! l

\Gamma

X

+

)

\Gamma (D

0

! l

+

X

\Gamma

)

= exp(\Gamma 4ImOE)

1 \Gamma ff

1 + ff

; (3a)

_r j

\Gamma (

_ D

0

! l

+

X

\Gamma

)

\Gamma (

_ D

0

! l

\Gamma

X

+

)

= exp(+4ImOE)

1 \Gamma ff

1 + ff

; (3b)

where ff = (1 \Gamma y

2

D

)=(1 + x

2

D

). Note that nonvanishing ImOE signifies CP violation in D

0

\Gamma

_ D

0

mixing. To fit more accurate data in the near future, we prefer the following mixing parameter:

r

D

j

r + _r

2

= cosh(4ImOE)

1 \Gamma ff

1 + ff

: (4)

For ImOE , 1%, the value of cosh(4ImOE) deviates less than 0:1% from unity. Thus this overall factor of r

D

is safely negligible

3

. The latest E691 data [6] give r ss _r ss r

D

ss (x

2

D

+ y

2

D

)=2 !

0:37% for small x

D

and y

D

, where exp(\Gamma 4ImOE) ss exp(+4ImOE) ss 1, a worse approximation

than cosh(4ImOE) ss 1, has been used.

For a o/ -charm factory running at the (4:14) resonance, the coherent D

0

_

D

0

events can be

produced through (4:14) ! fl(D

0

_

D

0

)

C=+

or (4:14) ! ss

0

(D

0

_

D

0

)

C=\Gamma

, where C stands for

the charge-conjugation parity [7, 13]. 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

; (5)

where k is the three-momentum vector of D

0

and

_ D

0

mesons. For our purpose, we only

consider the primary dilepton events which are directly emitted from the coherent (D

0

_

D

0

)

C

decays. Let N

\Sigma \Sigma

C

and N

+\Gamma

C

denote the time-integrated numbers of like-sign and opposite-sign

dilepton events, respectively. After a straightforward calculation [7, 14, 15], we obtain

N

++

C

= N

C

exp(+4ImOE)

"

1 + Cy

2

D

(1 \Gamma y

2

D

)

2

\Gamma

1 \Gamma Cx

2

D

(1 + x

2

D

)

2

#

; (6a)

3

Indeed the magnitude of ImOE can be determined by measurements of the CP asymmetry (_r \Gamma r)=(_r + r),

which is equal to sinh(4ImOE)= cosh(4ImOE) ss 4ImOE.

3

N

\Gamma \Gamma

C

= N

C

exp(\Gamma 4ImOE)

"

1 + Cy

2

D

(1 \Gamma y

2

D

)

2

\Gamma

1 \Gamma Cx

2

D

(1 + x

2

D

)

2

#

; (6b)

and

N

+\Gamma

C

= 2N

C

"

1 + Cy

2

D

(1 \Gamma y

2

D

)

2

+

1 \Gamma Cx

2

D

(1 + x

2

D

)

2

#

; (7)

where N

C

is the normalization factor proportional to the rates of semileptonic D

0

and

_ D

0

decays. It is easy to check that the relation

N

++

\Gamma

N

\Gamma \Gamma

+

= N

++

+

N

\Gamma \Gamma

\Gamma

(8)

holds stringently, and it is independent of the magnitudes of D

0

\Gamma

_ D

0

mixing and CP violation.

Note that a coherent D

0

_

D

0

pair with C = \Gamma can be straightforwardly produced from the

decay of the (3:77) resonance [10, 13]. Its time-independent decay rates to the like-sign and opposite-sign dileptons obey eqs. (6) and (7), respectively. At a o/ -charm factory the (D

0

_

D

0

)

C=\Gamma

events at both the (3:77) and (4:14) resonances will be accumulated, and a combination of them may increase the sensitiveness of our approach to probing D

0

\Gamma

_ D

0

mixing.

Usually one is interested in the following two types of observables:

a

C

j

N

++

C

\Gamma N

\Gamma \Gamma

C

N

++

C

+ N

\Gamma \Gamma

C

; r

C

j

N

++

C

+ N

\Gamma \Gamma

C

N

+\Gamma

C

; (9)

which signify nonvanishing CP violation and D

0

\Gamma

_ D

0

mixing, respectively. Explicitly, we find

a

\Gamma

= a

+

=

sinh(4ImOE)

cosh(4ImOE)

ss 4ImOE (10)

for small ImOE. If ImOE is at the level of 10

\Gamma 3

or larger, it can be measured to 3 standard deviations

at the second-round experiments of a o/ -charm factory with about 10

7

like-sign dileptons (or

equivalently, about 10

10

D

0

_

D

0

events). Furthermore,

r

\Gamma

= cosh(4ImOE)

1 \Gamma ff

1 + ff

; r

+

= cosh(4ImOE)

fi \Gamma ff

2

fi + ff

2

; (11)

where fi = (1 + y

2

D

)=(1 \Gamma x

2

D

). One can see that r

\Gamma

= r

D

holds without any approximation.

For small x

D

and y

D

, we have r

\Gamma

ss (x

2

D

+ y

2

D

)=2 and r

+

ss 3r

\Gamma

. These two approximate results

are well-known in the literature (see, e.g., [7]). In such an approximation, however, the relative size of x

2

D

and y

2

D

cannot be determined.

To distinguish between the different contributions of x

D

and y

D

to D

0

\Gamma

_ D

0

mixing, one has

to measure r

\Sigma

as precisely as possible. With the help of eq. (11), we show that the magnitudes

4

of x

D

and y

D

can be separately determined as follows:

x

2

D

=



1 + r

\Gamma

1 \Gamma r

\Gamma

\Delta

1 + 3r

\Gamma

1 \Gamma r

\Gamma

\Gamma

1 + r

+

1 \Gamma r

+

!

1 + r

\Gamma

1 \Gamma r

\Gamma

\Gamma

1 + r

+

1 \Gamma r

+

!

\Gamma 1

; (12a)

y

2

D

=



1 \Gamma r

\Gamma

1 + r

\Gamma

\Delta

1 \Gamma 3r

\Gamma

1 + r

\Gamma

\Gamma

1 \Gamma r

+

1 + r

+

!

1 \Gamma r

+

1 + r

+

\Gamma

1 \Gamma r

\Gamma

1 + r

\Gamma

!

\Gamma 1

: (12b)

Here it is worth emphasizing that cosh(4ImOE) as the overall (and common) factor of r

D

, r

\Gamma

and r

+

can be safely neglected. In the approximations up to O(r

2

\Gamma

) and O(r

2

+

), we obtain two

simpler relations:

x

2

D

\Gamma y

2

D

ss 2

r

+

\Gamma 3r

\Gamma

r

+

\Gamma r

\Gamma

; x

2

D

+ y

2

D

ss 4r

\Gamma

r

+

\Gamma 2r

\Gamma

r

+

\Gamma r

\Gamma

: (13)

Thus it is crucial to examine the deviation of the ratio r

+

=r

\Gamma

from 3, in order to find the

difference between x

2

D

and y

2

D

. Instructively, we consider three special cases:

x

D

?? y

D

=)

r

+

r

\Gamma

ss 3 + 2r

\Gamma

? 3 ; (14a)

x

D

ss y

D

=)

r

+

r

\Gamma

ss 3 \Gamma 9r

2

\Gamma

ss 3 ; (14b)

x

D

!! y

D

=)

r

+

r

\Gamma

ss 3 \Gamma 2r

\Gamma

! 3 : (14c)

These relations can be directly derived from eq. (11) or (12). If r

\Gamma

is close to the current

experimental bound (i.e., r

\Gamma

= r

D

ss (x

2

D

+ y

2

D

)=2 ! 0:37%), then measurements of r

+

=r

\Gamma

to the accuracy of 10

\Gamma 4

can definitely establish the relative magnitude of x

D

and y

D

. To

this goal, about 10

8

like-sign dileptons (or equivalently, about 10

11

events of (D

0

_

D

0

)

C=\Gamma

and

(D

0

_

D

0

)

C=+

pairs) are needed. Certainly such a measurement can only be carried out at the

second-generation o/ -charm factories (beyond the ones under consideration at present).

In principle, there is another possibility to determine x

2

D

and y

2

D

separately. If the produc-

tion cross-sections of (4:14) ! fl(D

0

_

D

0

)

C=+

and (4:14) ! ss

0

(D

0

_

D

0

)

C=\Gamma

can be reliably

predicted or measured, it is possible to fix the ratio of the two normalization factors (i.e., n j N

\Gamma

=N

+

) independent of the dilepton events shown in eqs. (6) and (7). Then an asymme-

try between the C = \Gamma and C = + opposite-sign dilepton events can be obtained as follows:

\Delta

+\Gamma

j

N

+\Gamma

+

\Gamma N

+\Gamma

\Gamma

N

+\Gamma

+

+ N

+\Gamma

\Gamma

ss

1 \Gamma n

1 + n

\Gamma

3 \Gamma n

1 + n

\Delta

x

2

D

\Gamma y

2

D

2

: (15)

We see that n = 1 is the most favorable case (i.e., (D

0

_

D

0

)

C=+

and (D

0

_

D

0

)

C=\Gamma

events have the

same production rate at the (4:14) resonance), in which \Delta

+\Gamma

directly measures the difference

5

between x

2

D

and y

2

D

. A comparison between \Delta

+\Gamma

and r

\Gamma

(or r

+

) is able to determine x

2

D

and y

2

D

separately. In this way r

\Gamma

need not be measured as precisely as in the first approach discussed

above, however, the accurate value of n is necessary. Unfortunately, it seems impossible at present to precisely determine n from either theory or experiments.

We have shown that it is possible to separately determine the D

0

\Gamma

_ D

0

mixing parameters

x

D

and y

D

by time-integrated measurements of the dilepton events of (D

0

_

D

0

)

C=\Sigma

decays at

a o/ -charm factory. In the assumption of a dedicated accelerator running for one year at an average luminosity of 10

33

s

\Gamma 1

cm

\Gamma 2

, about 10

7

events of fl(D

0

_

D

0

)

C=+

and the similar number

of ss

0

(D

0

_

D

0

)

C=\Gamma

are expected to be produced at the (4:14) resonance [13]

4

. The precision

of 10

\Gamma 4

to 10

\Gamma 5

in measurements of r

\Gamma

and r

+

is achievable if one assumes zero background

and enough running time [13, 16]. To measure the ratio r

+

=r

\Gamma

up to the accuracy of 10

\Gamma 4

,

however, much experimental effort is needed. If D

0

\Gamma

_ D

0

mixing were at the level of r

D

, 10

\Gamma 3

(or at least r

D

* 10

\Gamma 4

), then the relative magnitude of x

D

and y

D

should be detectable in the

second-round experiments of a o/ -charm factory.

I would like to thank H. Fritzsch and A.I. Sanda for their warm hospitality during my research stay in M"unchen and Nagoya, respectively. I am also grateful to D. M. Kaplan and T. Liu for some useful discussions. This work was supported by both the Alexander von Humboldt Foundation and the Japan Society for the Promotion of Science.

References

[1] H.Y. Cheng, Phys. Rev. D26 (1982) 143; A. Datta and D. Kumbhakar, Z. Phys. C27 (1985)

515.

[2] J.F. Donoghue et al., Phys. Rev. D33 (1986) 179. [3] L. Wolfenstein, Phys. Lett. B164 (1985) 170. [4] H. Georgi, Phys. Lett. B297 (1992) 353; T. Ohl, G. Ricciardi, and E.H. Simmons, Nucl.

Phys. B403 (1993) 605. 4

A rough estimate gives n ss 0:78. Note that more (D

0

_

D

0

)

C=\Gamma

events can be produced at the (3:77)

resonance, but they are only applicable to the measurement of r

\Gamma

.

6

[5] For a brief review, see: G. Burdman, Report No. Fermilab-Conf-94/200 (1994); Fermilab-

Conf-95/281-T (1995).

[6] Particle Data Group, L. Montanet et al., Phys. Rev. D50 (1994) 1173. [7] I.I. Bigi and A.I. Sanda, Phys. Lett. B171 (1986) 320; I.I. Bigi, SLAC Report No. 343

(1989) 169; I.I. Bigi, Report No. Fermilab-Conf-94/190 (1994).

[8] L. Wolfenstein, Phys. Rev. Lett. 75 (1995) 2460. [9] S. Pakvasa, Report No. UH-511-787-94 (1994); J.L. Hewett, Report No. SLAC-PUB-95-

6821 (1995); G. Blaylock, A. Seiden, and Y. Nir, Phys. Lett. B355 (1995) 555; T.E. Browder and S. Pakvasa, Report No. UH-511-828-95 (1995).

[10] Z.Z. Xing, Report No. LMU-20/95 (accepted for publication in Phys. Lett. B). [11] D.M. Kaplan, Report No. Fermilab-Conf-94/190 (1994); IIT-HEP-95/3 (1995). [12] T. Liu, Report No. HUTP-94/E021 (1994); Princeton/HEP/95-6 (1995). [13] J.R. Fry and T. Ruf, Report No. CERN-PPE/94-20 (1994). [14] Explicit formulas for coherent B

0

_

B

0

decays can be found in: I.I. Bigi and A.I. Sanda,

Nucl. Phys. B281 (1987) 41; I.I. Bigi, V.A. Khoze, N.G. Uraltsev, and A.I. Sanda, in CP Violation, edited by C. Jarlskog (World Scientific, Singapore, 1988), p. 175.

[15] A generic formalism for the time-dependent and time-integrated decays of coherent P

0

_

P

0

pairs (P = K; D; B

0

d

or B

0

s

) has been given by Z.Z. Xing, in Phys. Rev. D53 (1996) 204.

[16] See, e.g., G. Gladding, SLAC Report No. 343 (1989) 159; U. Karshon, SLAC Report No.

343 (1989) 706; W. Toki, SLAC Report No. 343 (1989) 57.

7

