Testing CPT invariance in B0d– �B0

d and B0s– �B0

s oscillations

Ping Ren and Zhi-zhong Xing*Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918, Beijing 100049, China

(Received 26 August 2007; published 5 December 2007)

Recent CDF and D0 measurements of B0s– �B0

s mixing make it possible to search for CP violation andtest CPT symmetry in a variety of Bs decays. Considering both coherent B0

d�B0d decays at the ��4S�

resonance and coherent B0s

�B0s decays at the ��5S� resonance, we formulate their time-dependent and time-

integrated rates by postulating small CPT violation in B0d– �B0

d and B0s– �B0

s oscillations. We show that theopposite-sign dilepton events from either C-odd or C-even B0

q�B0q states (for q � d or s) can be used to

determine or constrain the CPT-violating parameter at a super-B factory. The possibility of distinguishingbetween the effect of CPT violation and that of �B � ��Q transitions is also discussed.

DOI: 10.1103/PhysRevD.76.116001 PACS numbers: 11.30.Er, 13.25.Hw, 14.40.Nd

I. INTRODUCTION

A correlated P0 �P0 system, where P may be either K, D,Bd, or Bs, has been of great interest for the study of CP, T,and CPT symmetries in particle physics. In the K0– �K0

mixing system, for instance, both indirect CP violation [1]and direct CP violation [2] have unambiguously beenobserved; the evidence for T violation [3] has beenachieved; and CPT invariance has been tested to an im-pressive degree of accuracy [4]. The �Q � �S rule hasalso been examined in the semileptonic K0 and �K0 tran-sitions. Beyond the K0 �K0 system, both indirect and directsignals of CP violation have been observed in a number ofneutral Bd decays [4]; and possible CPT violation inB0d– �B0

d mixing has been searched for at the KEK andSLAC B-meson factories [5]. Although the phenomenaof CP violation have not been seen in the B0

s– �B0s and

D0– �D0 mixing systems, they may show up and even sur-prise us in the near future at the LHC-B [6], �-charm [7],and super-B [8] factories.

The CDF [9] and D0 [10] collaborations have recentlyreported their measurements of B0

s– �B0s mixing (both the

mass and width differences between the light and heavy Bsmass eigenstates) at the Fermilab Tevatron Collider. Thefirst measurement is from the CDF collaboration and thesecond measurement is from the D0 collaboration:

�Ms � 17:77� 0:10�stat� � 0:07�syst� ps�1;

��s � 0:13� 0:09 ps�1:(1)

This remarkable progress in experimental B physics makesit possible to search for CP violation and test CPT sym-metry in the B0

s�B0s system. As the magnitude of B0

s– �B0s

mixing is much larger than that of B0d– �B0

d mixing [4], itsimpact on the time-dependent and time-integrated rates ofB0s and �B0

s decays deserves attention. In particular, the CP-and CPT-violating signals might get enhanced or sup-pressed due to large B0

s– �B0s mixing. At a super-B factory

with the luminosity L� a few� 1036 cm�2 s�1 [8], bothcoherent B0

d�B0d decays at the ��4S� resonance and coherent

B0s

�B0s decays at the ��5S� resonance will be studied down

to the last detail.The main purpose of this work is to formulate the decay

rates of a correlated B0q

�B0q state (either q � d or q � s) in

the assumption that there are both small CPT violation inB0q– �B0

q oscillation and small �B � �Q violation in semi-leptonic B0

q and �B0q decays. Note that possible effects of

CPT violation in neutral Bd and D decays have beenanalyzed in Refs. [11–15], respectively; and possible ef-fects of �B � ��Q and �C � ��Q transitions havebeen discussed in Refs. [16,17], respectively. The presentpaper is different from the previous ones not only becausewe are dealing with a new heavy meson-antimeson mixingsystem (i.e., the B0

s�B0s system) but also because our dis-

cussions are essentially new in two aspects. (1) We shalltake into account both C-odd and C-even B0

q�B0q pairs with

C being the charge-conjugation parity of this coherentsystem, and calculate both time-dependent and time-integrated rates of �B0

q�B0q�C decays by assuming slight

CPT violation and small �B � ��Q effects. Our analyti-cal results are more general and more useful than thoseobtained in Refs. [11–16]. (2) We shall concentrate on theopposite-sign dilepton asymmetries of �B0

q�B0q�C decays to

investigate possible effects of CPT violation and �B ���Q transitions. It is worth remarking that an opposite-sign dilepton event may be either l�1 l

�2 (for l1 � l2) or

l�l�. Although the opposite-sign dilepton events of�B0

d�B0d�C decays have been considered in Refs. [12,13],

our results for the B0s– �B0

s mixing system have their ownfeatures and implications due to the small CP-violatingphase of B0

s– �B0s mixing and the large values of �Ms and

��s.So far some interest has been paid to the possibilities of

exploring CP violation and probing new physics in weakBs decays at the ��5S� resonance [18–21], although theexperimental feasibility remains an open question. Weexpect that the future super-B factory can run at thisinteresting energy threshold [22]. Then it will be possible*xingzz@ihep.ac.cn

PHYSICAL REVIEW D 76, 116001 (2007)

1550-7998=2007=76(11)=116001(6) 116001-1 © 2007 The American Physical Society

to test CPT invariance in both B0d– �B0

d and B0s– �B0

s oscilla-tions by studying the coherent B0

d�B0d and B0

s�B0s decays.

II. CPT VIOLATION IN COHERENT �B0q

�B0q�C

DECAYS

The mixing or oscillation between B0q and �B0

q mesonscan naturally arise from their common coupling to a subsetof real and virtual intermediate states. Hence the masseigenstates jBLi and jBHi, where ‘‘L’’ (‘‘H’’) denotes‘‘light’’ (‘‘heavy’’), are different from the flavor (weakinteraction) eigenstates jB0

qi and j �B0qi. Taking account of

both CP- and CPT-violating effects in B0q- �B0

q mixing, onemay parametrize the correlation between fjBLi; jBHig andfjB0

qi; j �B0qig states as follows:

jBLi � cos�2e�i��=2�jB0

qi � sin�2e�i��=2�j �B0

qi;

jBHi � sin�2e�i��=2�jB0

qi � cos�2e�i��=2�j �B0

qi;

(2)

where � and � are in general complex. For simplicity, thenormalization factors of jBLi and jBHi in Eq. (2) have beenomitted. CPT invariance requires cos� � 0 or equivalently� � �=2, while CP conservation requires both � � �=2and � � 0 [23].1 The proper-time evolution of an initiallypure jB0

qi or j �B0qi state is given by [11]

jB0q�t�i � e��iM���=2��tg��t�jB0

qi � ~g��t�j �B0qi;

j �B0q�t�i � e��iM���=2��tg��t�j �B0

qi � ~g��t�jB0qi;

(3)

where

g��t� � cosh�ixq � yq

2�t�� cos� sinh

�ixq � yq2

�t�;

~g��t� � sin�e�i� sinh�ixq � yq

2�t�: (4)

The definitions in Eqs. (3) and (4) are M � �ML �MH�=2,� � ��L � �H�=2, xq � �Mq=� with �Mq � MH �ML,and yq � ��q=�2�� with ��q � �L � �H, where ML;H

(�L;H) denotes the mass (width) of BL;H. Taking accountof 1=� � 1:52 ps [10], we approximately obtain xs � 27and ys � 0:1 from the central values of �Ms and ��sgiven in Eq. (1). In contrast, xd � 0:78 [4] has been known

but yd has not been measured for the B0d– �B0

d mixing sys-tem. We stress that the experimental values of xd, xs and ysare in good agreement with the standard-model predictions[27],2 although the uncertainty associated with ys remainsquite large.

In order to calculate the proper-time distribution ofcoherent �B0

q�B0q�C decays, we neglect the tiny final-state

electromagnetic interactions and assume CPT invariancein the direct transition amplitudes of semileptonic or non-leptonic B0

q and �B0q decays. Such an assumption can be

examined, without the mixing-induced complexity, by de-tecting the charge asymmetry of semileptonic B� decays[12]. We shall take into account possible �B � ��Qtransitions in our calculations. The latter can be describedby using a small parameter �l for a given semileptonicdecay mode,

hl�X�l jH jB0qi � Al; hl�X�l jH j �B

0qi � �lAl;

hl�X�l jH j �B0qi � A l ; hl�X�l jH jB

0qi � � l A

l ;

(5)

where �l measures the �B � ��Q effect and j�lj � 1 isexpected to hold. j�lj � 0 implies that it is in practiceimpossible to have a pure tagging of the B0

q or �B0q state

through its semileptonic decay (to l�X�l or l�X�l ). Ingeneral, the amplitudes of B0

q and �B0q decays into the

final-state fi (either semileptonic or nonleptonic) are de-noted as Afi � hfijH jB

0qi and �Afi � hfijH j

�B0qi. When

the coherent �B0q

�B0q�C ! f1f2 decays are concerned, the

following combinations

�C � e�i� � Cei��Af1

�Af2

Af1Af2

; �C ��Af2

Af2

� C�Af1

Af1

; (6)

where C � �1, will be frequently used.Now let us consider a correlated B0

q�B0q state at rest. Its

time-dependent wave function can be written as

1���2p jB0

q�K; t�ij �B0q��K; t�i � CjB0

q��K; t�ij �B0q�K; t�i;

(7)

where K is the three-momentum vector of B0q and �B0

q, andC � �1 is the charge-conjugation parity of this coherentsystem. If one of the two Bq mesons (with momentum K)decays to a final state f1 at proper time t1 and the other(with�K) to f2 at t2, the amplitude of their joint decays isgiven by

1As CPT violation may simultaneously imply the violation ofLorentz covariance in a quantum field theory [24], the depen-dence of � on the sidereal time should in general be taken intoaccount [25]. For simplicity, here we take � as a constant byassuming that the boost parameters of B0

q and �B0q mesons are

small and the corresponding Lorentz violation is rotationallyinvariant in the laboratory frame. In this approximation, ourresults are valid as averages over the sidereal time, such thatthe effect of Lorentz violation due to the direction of motion canbe neglected. A complete analysis, which requires incorporatingLorentz-violating parameters directly into the phenomenology toaccount for possible CPT violation [26], is beyond the scope ofthis paper and will be done elsewhere.

2Note that yd � 0:002 and ys � 0:06 � � � 0:08 are the updatedpredictions [27]. The theoretical expectation xs=xd � ys=yd �35 has partly been confirmed by current experimental data.

PING REN AND ZHI-ZHONG XING PHYSICAL REVIEW D 76, 116001 (2007)

116001-2

A�f1; t1; f2; t2�C �1���2p e��iM���=2���t1�t2�fAf1

Af2g��t1�~g��t2� � C~g��t1�g��t2� � Af1

�Af2g��t1�g��t2�

� C~g��t1�~g��t2� � �Af1Af2~g��t1�~g��t2� � Cg��t1�g��t2� � �Af1

�Af2~g��t1�g��t2�

� Cg��t1�~g��t2�g: (8)

The calculation of the decay rate R�f1; t1; f2; t2�C / jA�f1; t1; f2; t2�Cj2 is straightforward but lengthy. Our result is

R�f1; t1; f2; t2�C / jAf1j2jAf2

j2e��t��j�Cj2 � j�Cj

2� cosh�yq�tC� � �j�Cj2 � j�Cj

2� cos�xq�tC�

� 2 Re�� C�C� sinh�yq�tC� � 2 Im�� C�C� sin�xq�tC� �V �f1; t1; f2; t2�C; (9)

where tC � t2 � Ct1 is defined, �C and �C have been given in Eq. (6), and V �f1; t1; f2; t2�C denotes the CPT-violatingterm:

V �f1; t1; f2; t2�� � �2 Re���� � � � ���� cos� cosh�yq�t�� � 2 Re���� � � � ���� cos� cos�xq�t��

� 2 Re�j��j2 � � ���� cos� sinh�yq�t�� � 2 Im�j��j2 � � ���� cos� sin�xq�t��

� j����� � ���jj cos�je�yq�t1 cos�xq�t2 ���� � j����� � ���jj cos�je�yq�t1 cos�xq�t2 ����

� j����� � ���jj cos�je�yq�t2 cos�xq�t1 ���� � j����� � ���jj cos�je�yq�t2 cos�xq�t1 ����

(10)

with

tan�� �Im����

� � �

�� cos�

Re���� � � �

�� cos�

; (11)

or

V �f1; t1; f2; t2�� � �2 Re���� � cos�� cosh�yq�t�� � 2 Re���� � cos�� cos�xq�t�� � 2 Re���� � cos�� sinh�yq�t��

� 2 Im���� � cos�� sin�xq�t�� � j��� � ������ � ���jj cos�je�yq�t1 cos�xq�t2 �����

� j��� � ������ � ���jj cos�je�yq�t1 cos�xq�t2 ����� � j��� � ������ � ���jj cos�je�yq�t2

� cos�xq�t1 ����� � j��� � ������ � ���jj cos�je�yq�t2 cos�xq�t1 ����� (12)

with

tan��� �Im�� � � �

����� � ��� cos�

Re�� � � � ����� � ��� cos�

: (13)

We observe that the CPT-violating term has a more complicated time-dependent behavior than the CPT-conserving term.For completeness, the time-integrated form of R�f1; t1; f2; t2�C is given by

R�f1; f2�C / jAf1j2jAf2

j2�

1� Cy2q

�1� y2q�

2 �j�Cj2 � j�Cj

2� �1� Cx2

q

�1� x2q�

2 �j�Cj2 � j�Cj

2� �2�1� C�yq�1� y2

q�2 Re�� C�C�

�2�1� C�xq�1� x2

q�2 Im�� C�C� � 2V �f1; f2�C

�; (14)

where the CPT-violating term reads

V �f1; f2�� � �1

1� y2q

Re���� � � � ���� cos� �1

1� x2q

Re���� � � � ���� cos�

�2

�1� x2q��1� y2

q�Re���� � cos�� � xqyq Im���� � cos��; (15)

or

TESTING CPT INVARIANCE IN . . . PHYSICAL REVIEW D 76, 116001 (2007)

116001-3

V �f1; f2�� � �

�1� y2

q

�1� y2q�

2 �1� x2

q

�1� x2q�

2

�Re����

� cos�� �

2yq�1� y2

q�2 �x2q � y2

q

1� x2q

Re���� � cos��

�2xq

�1� x2q�

2 �x2q � y

2q

1� y2q

Im���� � cos�� �

2

�1� x2q��1� y2

q�Re����

� cos�� � xqyq Im����

� cos��: (16)

When CPT is a good symmetry ( cos� � 0), Eqs. (9) and(14) are in agreement with the formulas obtained inRefs. [15,28].

We stress that Eqs. (9)–(16) are new results and mayserve as the master equations for the study of CPT viola-tion in coherent �B0

q�B0q�C decays. They are also valid for

other correlated meson-antimeson systems with CPT vio-lation, in particular, useful to describe coherent �D0 �D0�Cdecays at the �3770� and �4140� resonances.

III. EXAMPLE: OPPOSITE-SIGN DILEPTONEVENTS

To be specific, we consider a particularly simple andinteresting possibility of testing CPT invariance in B0

q– �B0q

mixing: the opposite-sign dilepton events from coherent�B0

q�B0q�C decays. One may take f1 � l�1 X

�1 and f2 � l�2 X

�2

with either l1 � l2 (e.g., l1 � l2 � �) or l1 � l2 (e.g., l1 �e and l2 � �). The amplitude of each semileptonic decaymode can be parameterized in analogy with Eq. (5). Sincethe �B � ��Q transitions must be strongly suppressed, itis reasonable to take j�li j � 1 (for i � 1 or 2) in ourcalculations.

We first look at the C � �1 case. By simplifyingEqs. (9)–(11), we obtain the time-dependent rates of�B0

q�B0q�� ! �l�1 X

�1 �t1�l

�2 X�2 �t2 decays as follows:

R�l�1 X�1 ; t1; l�2 X

�2 ; t2�� / jAl1 j

2jAl2 j2e��t�cosh�yq�t�� � cos�xq�t�� � 2 Re� sinh�yq�t�� � 2 Im� sin�xq�t��;

R�l�1 X�1 ; t1; l�2 X

�2 ; t2�� / jAl1 j

2jAl2 j2e��t�cosh�yq�t�� � cos�xq�t�� � 2 Re �� sinh�yq�t�� � 2 Im �� sin�xq�t��;

(17)

where t� � t2 � t1, and

� � cos�� �l1e�i� � � l2e

�i�;

�� � cos�� � l1e�i� � �l2e

�i�(18)

have been defined by keeping the leading terms of CPTviolation and �B � ��Q effects. Note that ei� ��V tbVts�=�VtbV

ts� � 1� 2i2 holds for B0

s– �B0s mixing

described by the box diagrams in the standard model,where � 0:22 and � 0:34 are the Wolfensteinparameters [4]. If e�i� � 1 is taken in the leading-orderapproximation, Eq. (18) can be simplified to � � cos����l1 � �

l2� and �� � cos�� �� l1 � �l2�. If l1 � l2 is fur-

ther taken, then we have � � �� � cos�� 2i Im��l1�.It is remarkable that the same simplification cannot bemade for the B0

d– �B0d mixing system, where ei� �

�V tbVtd�=�VtbV td� � e�2i� with � � 22� being one of the

inner angles of the Cabibbo-Kobayashi-Maskawa unitaritytriangle in the standard phase convention [4]. Of course,ei� might deviate from the standard-model expectation ifB0q– �B0

q mixing (for q � d or s) involves a kind of new

physics. The latter has also been included into the parame-ters � and ��. Thus these two parameters serve for aneffective description of possible new physics (CPT viola-tion, �B � ��Q transitions, and new �B � 2 effects) inB0q– �B0

q mixing.Equations (17) and (18) clearly show that the �B �

��Q parameters have the same time-dependent behavioras the CPT-violating parameter cos� in the opposite-signdilepton events of the correlated B0

q�B0q state with C � �1.

Hence it is in general impossible to distinguish between theeffect of CPT violation and that of �B � ��Q transitionsin this kind of events, unless one of them is remarkablysmaller than the other. If the decays of the correlated B0

q�B0q

state with C � �1 are taken into account, however, it is inprinciple possible to cleanly extract the CPT-violatingparameter [13]. To illustrate this point in a transparentway, we simplify Eqs. (9), (12), and (13) to obtain thetime-dependent rates of �B0

q�B0q�� ! �l

�1 X�1 �t1�l

�2 X�2 �t2 de-

cays. The result is

R�l�1 X�1 ; t1; l�2 X

�2 ; t2�� / jAl1 j

2jAl2 j2e��t�cosh�yq�t�� � cos�xq�t�� � 2 Re�0 sinh�yq�t�� � 2 Im�0 sin�xq�t��

� 2��t1; t2�;

R�l�1 X�1 ; t1; l�2 X

�2 ; t2�� / jAl1 j

2jAl2 j2e��t�cosh�yq�t�� � cos�xq�t�� � 2 Re ��0 sinh�yq�t�� � 2 Im ��0 sin�xq�t��

� 2��t1; t2�;

(19)

PING REN AND ZHI-ZHONG XING PHYSICAL REVIEW D 76, 116001 (2007)

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where �0 � �l1e�i� � � l2e

�i� and ��0 � � l1e�i� � �l2e

�i� do not contain the CPT-violating effect, but ��t1; t2� ispurely a CPT-violating term:

��t1; t2� � �cos�xq�t1� sinh�yq�t2� � sinh�yq�t1� cos�xq�t2�Re�cos��

� sin�xq�t1� cosh�yq�t2� � cosh�yq�t1� sin�xq�t2� Im�cos��: (20)

One can easily see that ��t2; t1� � ���t1; t2� holds, but the CPT-conserving terms in Eq. (19) do not change with theexchange of t1 and t2. This interesting feature implies that ��t1; t2� can in principle be extracted from the rate differences

R�l�1 X�1 ; t1; l�2 X

�2 ; t2�� � R�l

�1 X�1 ; t2; l�2 X

�2 ; t1�� / 4jAl1 j

2jAl2 j2e��t���t1; t2�;

R�l�1 X�1 ; t2; l�2 X

�2 ; t1�� � R�l

�1 X�1 ; t1; l�2 X

�2 ; t2�� / 4jAl1 j

2jAl2 j2e��t���t1; t2�:

(21)

As Im� � 0 is a good approximation for B0q– �B0

q mixing in the standard model, we have Re ��0 � Re�0 and Im ��0 ��Im�0. In this case, Im�0 can in principle be extracted from the rate differences

R�l�1 X�1 ; t1; l�2 X

�2 ; t2�� � R�l

�1 X�1 ; t2; l�2 X

�2 ; t1�� / 4jAl1 j

2jAl2 j2e��t� Im�0;

R�l�1 X�1 ; t2; l�2 X

�2 ; t1�� � R�l

�1 X�1 ; t1; l�2 X

�2 ; t2�� / 4jAl1 j

2jAl2 j2e��t� Im�0:

(22)

For B0s– �B0

s mixing with e�i� � 1, Im�0 � Im�l1 � Im�l2holds; and for B0

d– �B0d mixing with e�i� � e�2i�, we

obtain Im�0 � �Re�l2 � Re�l1� sin2�� �Im�l2 �Im�l1� cos2�. Thus the dilepton events of coherent�B0

q�B0q�� decays are very useful to probe possible CPT

violation and �B � ��Q effects.If the forthcoming super-B factory is also an asymmetric

e�e� collider as the present KEK and SLAC B-mesonfactories, it will be easier to measure the proper-timedifference t� � �t2 � t1� of a dilepton event. A measure-ment of the t� � �t2 � t1� distribution might be difficult ineither linacs or storage rings, unless the bunch lengths aremuch shorter than the decay lengths [29]. Hence we may

calculate the t� distribution of the dilepton events byintegrating R�l�1 X

�1 ; t1; l�2 X

�2 ; t2�C over t�. For simplicity,

here we assume �B � �Q to be a perfect rule and use t todenote t�. We take t > 0 by convention. Our results are

R�l�1 X�1 ; l

�2 X�2 ; t�� / jAl1 j

2jAl2 j2e��tcosh�yq�t�

� cos�xq�t� � 2 Re�cos��

� sinh�yq�t� � 2 Im�cos��

� sin�xq�t�; (23)

and

R�l�1 X�1 ; l

�2 X�2 ; t�� / jAl1 j

2jAl2 j2e��t

�cosh�yq�t� ’y���������������

1� y2q

q �cos�xq�t� ’x���������������

1� x2q

q

�2j cos�jcos���!��e

�yq�t � cos���!� � xq�t��������������������������������x2q � �2� yq�2

q

�2j cos�jcos���!��e

�yq�t � cos���!� � xq�t��������������������������������x2q � �2� yq�2

q�; (24)

where the parameters ’x, ’y, !�, and � are definedthrough tan’x � xq, tanh’y � yq, tan!� � xq=�2� yq�,and tan� � Im�cos��=Re�cos��, respectively. Taking xs �27 and ys � 0:07 for example, we have ’x � 1:53, ’y �0:07, !� � 1:49, and !� � 1:50; while taking xd � 0:78and yd � 0:002 for example, we have ’x � 0:66, ’y �0:002, and !� � !� � 0:37. Equations (23) and (24)show that both Re�cos�� and Im�cos�� can in principlebe determined or constrained by measuring the decay ratesR�l�1 X

�1 ; t1; l�2 X

�2 ; t2�C, provided the �B � ��Q transi-

tions and other new-physics effects are negligibly small.

These formulas are also applicable for the D0– �D0 mixingsystem.

IV. SUMMARY

Keeping with the great experimental interest in testingdiscrete symmetries and conservation laws at the presentand future B-meson factories, we have reformulated thetime-dependent and time-integrated rates of coherent�B0

q�B0q�C decays (q � d or s) by assuming small CPT

violation in B0q– �B0

q oscillation. Our results are new and

TESTING CPT INVARIANCE IN . . . PHYSICAL REVIEW D 76, 116001 (2007)

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generic, and thus they can serve as the master formulas forthe analysis of possible CPT-violating effects in both theB0d– �B0

d mixing system at the ��4S� resonance and theB0s– �B0

s mixing system at the ��5S� resonance. Taking theopposite-sign dilepton events, for example, we have shownthat it is possible to separately determine or constrain theparameters of CPT violation and �B � ��Q transitionsby measuring their time distributions in the C � �1 case.In the C � �1 case, however, the CPT-violating and�B � ��Q effects have the same time-dependent behav-ior and are in general indistinguishable.

We expect that a stringent test of CPT symmetry and the�B � �Q rule will finally be realized at a super-B factorywith the luminosity L� a few� 1036 cm�2 s�1, whereother kinds of new physics may also be explored. The

prospect of such ambitious experiments is by no meansdim, indeed.

Finally let us mention the evidence for D0– �D0 mixingachieved in the BABAR [30] and Belle [31] experiments. Itturns out that the mixing parameter yD is at the 1% leveland jxDj< jyDj is expected to hold [32]. Therefore itseems possible to test CP, T, and CPT symmetries in thecharm system in the (far) future. As we have emphasizedbefore, the master formulas obtained in this paper can allbe used to describe coherent D0 �D0 decays with small CPTviolation and (or) small �C � ��Q effects.

ACKNOWLEDGMENTS

This work was supported in part by the National NaturalScience Foundation of China.

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