D0-anti-D0 Mixing and CP Violation in D0 vs anti-D0 to K*(+-) K(-+) Decays

TL;DR

This paper explores how recent evidence of D0-anti-D0 mixing can be used to measure mixing parameters and CP violation in specific D meson decays, using time-dependent and time-independent methods at charm factories.

Contribution

It proposes novel methods to determine or constrain D0-anti-D0 mixing parameters and CP violation effects from decay measurements at charm factories.

Findings

Mixing parameters x and y can be extracted from time-dependent decay data.

Strong phase difference δ can be constrained through decay measurements.

Potential to detect CP violation effects beyond the Standard Model.

Abstract

The noteworthy BaBar and Belle evidence for $D^0$-$\bar{D}^0$ mixing motivates us to study its impact on $D^0\to K^{*\pm} K^\mp$ decays and their CP-conjugate processes. We show that both the $D^0$-$\bar{D}^0$ mixing parameters ($x$ and $y$) and the strong phase difference between $\bar{D}^0\to K^{*\pm}K^\mp$ and $D^0\to K^{*\pm}K^\mp$ transitions ($\delta$) can be determined or constrained from the time-dependent measurements of these decay modes. On the $\psi (3770)$ and $\psi (4140)$ resonances at a $\tau$-charm factory, it is even possible to determine or constrain $x$, $y$ and $\delta$ from the time-independent measurements of coherent $(D^0\bar{D}^0) \to (K^{*\pm} K^\mp)(K^{*\pm} K^\mp)$ decays. If the CP-violating phase of $D^0$-$\bar{D}^0$ mixing is significant in a scenario beyond the standard model, it can also be extracted from the $K^{*\pm} K^\mp$ events.

Equations

$$y^{\prime }\cos \varphi$$

$$y_{CP}$$

$$\frac{{\bar{A}}_{K^{\ast +}{K}^{-}}}{A_{K^{\ast +}{K}^{-}}}=\frac{A_{K^{\ast -}{K}^{+}}}{{\bar{A}}_{K^{\ast -}{K}^{+}}}\equiv \rho e^{i\delta },$$

$$|D_1⟩$$

$$|D_2⟩$$

$${\lambda }_{K^{\ast +}{K}^{-}}$$

$${\bar{\lambda }}_{K^{\ast -}{K}^{+}}$$

$$\Gamma [D^0(t)\to K^{\ast +}{K}^{-}]$$

$$\Gamma [{\bar{D}}^0(t)\to K^{\ast -}{K}^{+}]$$

$$\Gamma [D^0(t)\to K^{\ast -}{K}^{+}]$$

$$\Gamma [{\bar{D}}^0(t)\to K^{\ast +}{K}^{-}]$$

$$x_{\pm }^{\prime }$$

$$y_{\pm }^{\prime }$$

$$\Gamma [D^0(t)\to K^{\ast +}{K}^{-}]$$

$$\Gamma [{\bar{D}}^0(t)\to K^{\ast -}{K}^{+}]$$

$$\Gamma [D^0(t)\to K^{\ast -}{K}^{+}]$$

$$\Gamma [{\bar{D}}^0(t)\to K^{\ast +}{K}^{-}]$$

$${\Gamma }_C^{++}$$

$${\Gamma }_C^{--}$$

$${\Gamma }_C^{+-}$$

$${\Gamma }_C^{++}$$

$${\Gamma }_C^{--}$$

$${\Gamma }_C^{+-}$$

$$\frac{{\Gamma }_{-}^{++}}{{\Gamma }_{-}^{+-}}$$

$$\frac{{\Gamma }_{-}^{--}}{{\Gamma }_{-}^{+-}}$$

$$\frac{{\Gamma }_{-}^{++}-{\Gamma }_{-}^{--}}{{\Gamma }_{-}^{+-}}\approx \frac{4r\left[\left(1-{\rho }^4\right)\Delta +{\rho }^2\sin \delta \sin \varphi \right]}{1-2{\rho }^2\cos \left(2\delta \right)+{\rho }^4},$$

$$\frac{{\Gamma }_{+}^{++}}{{\Gamma }_{+}^{+-}}\approx \frac{{\Gamma }_{+}^{--}}{{\Gamma }_{+}^{+-}}\approx \frac{4{\rho }^2}{1+2{\rho }^2\cos \left(2\delta \right)+{\rho }^4}$$

$$\frac{{\Gamma }_{+}^{++}-{\Gamma }_{+}^{--}}{{\Gamma }_{+}^{+-}}\approx \frac{8\rho \left[\Delta \cos \varphi \left(y_{+}^{\prime }-{\rho }^2{y}_{-}^{\prime }\right)+\sin \varphi \left(y_{-}^{\prime }-{\rho }^2{x}_{+}^{\prime }\right)\right]}{1+2{\rho }^2\cos \left(2\delta \right)+{\rho }^4}.$$

$$A_{K^{\ast +}{K}^{-}}$$

$$A_{{\bar{K}}^{\ast 0}{K}^0}$$

$$A_{K^{\ast +}{\bar{K}}^0}$$

$${\left|\frac{A_{K^{\ast +}{K}^{-}}}{A_{K^{\ast +}{\bar{K}}^0}}\right|}^2$$

$${\left|\frac{A_{{\bar{K}}^{\ast 0}{K}^0}}{A_{K^{\ast +}{\bar{K}}^0}}\right|}^2$$

$$z$$

$$\phi$$

$$B_{K^{\ast +}{\bar{K}}^0}$$

$$B_{K^{\ast +}{K}^{-}}$$

$$B_{{\bar{K}}^{\ast 0}{K}^0}$$

$$B_{{\bar{K}}^{\ast 0}{K}^{+}}$$

Full text

PACS number(s): 11.30.Er, 12.15.Ff, 13.20.Fc, 13.25.Ft

1   The BaBar [1] and Belle [2] experiments have recently provided us with some noteworthy evidence for $D^{0}$-$\bar{D}^{0}$ mixing, a quantum phenomenon similar to $K^{0}$-$\bar{K}^{0}$, $B^{0}_{d}$-$\bar{B}^{0}_{d}$ or $B^{0}_{s}$-$\bar{B}^{0}_{s}$ mixing. Both experiments indicate a non-vanishing width difference between the mass eigenstates $D_{1}$ and $D_{2}$ of $D^{0}$ and $\bar{D}^{0}$ mesons,

[TABLE]

where the values of $y^{\prime}\cos\phi$ and $y_{CP}$ are extracted from the decay modes $D^{0}\rightarrow\bar{D}^{0}\rightarrow K^{+}\pi^{-}$ versus $D^{0}\rightarrow K^{+}\pi^{-}$ [1] and $D^{0}\rightarrow\bar{D}^{0}\rightarrow K^{+}K^{-}$ and $\pi^{+}\pi^{-}$ versus $D^{0}\rightarrow K^{+}K^{-}$ and $\pi^{+}\pi^{-}$ [2], respectively. By linking $y^{\prime}$ and $y_{CP}$ to the $D^{0}$-$\bar{D}^{0}$ mixing parameters $x\equiv(M_{2}-M_{1})/\Gamma$ and $y\equiv(\Gamma_{2}-\Gamma_{1})/(2\Gamma)$ with $\Gamma=(\Gamma_{1}+\Gamma_{2})/2$ and $\Gamma_{1,2}$ being the width of $D_{1,2}$, Nir has pointed out that $|y|\sim 0.01$, $|x|<|y|$ and small or vanishingly small $CP$ violation are expected in the $D^{0}$-$\bar{D}^{0}$ mixing system within the standard model [3]. Some other authors have also discussed possible implications of the BaBar and Belle measurements of $D^{0}$-$\bar{D}^{0}$ mixing, either within or beyond the standard model [4]–[9].

Unfortunately, current theoretical calculations of $D^{0}$-$\bar{D}^{0}$ mixing involve large uncertainties because of the dominance of long-distance contributions [10]. In the standard model, the values of $x$ and $y$ are expected to be a second-order effect of the $SU(3)$ flavor symmetry breaking [11]: $x,y\sim\sin^{2}\theta_{\rm C}\times[SU(3)~{}{\rm breaking}]^{2}$, where $\theta_{\rm C}\approx 13^{\circ}$ denotes the Cabibbo angle. A very reliable prediction for the size of $SU(3)$ breaking has been lacking, although many attempts have been made [10, 12]. Hence these two $D^{0}$-$\bar{D}^{0}$ mixing parameters might be only of limited use in testing the standard model and searching for new physics. From an experimental point of view, however, it is always desirable to measure or constrain $x$ and $y$ as accurately as possible.

Motivated by the afore-mentioned positive results from the BaBar and Belle experiments, here we aim to investigate the impact of $D^{0}$-$\bar{D}^{0}$ mixing on $D^{0}\rightarrow K^{*\pm}K^{\mp}$ decays and their $CP$-conjugate processes. Because $K^{*+}K^{-}$ (or $K^{*-}K^{+}$) is not a $CP$ eigenstate, the amplitudes of $D^{0}$ and $\bar{D}^{0}$ decays into $K^{*+}K^{-}$ (or $K^{*-}K^{+}$) may have a significant strong phase difference $\delta$. In contrast, $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{+}K^{-}$ decays do not involve such a strong phase difference. We show that both the $D^{0}$-$\bar{D}^{0}$ mixing parameters ($x$ and $y$) and the strong phase difference ($\delta$) can be determined or constrained from the time-dependent measurements of $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$ decays. On the $\psi(3770)$ and $\psi(4140)$ resonances at a $\tau$-charm factory (e.g., BEPC-II [13]), we find that it is even possible to determine or constrain $x$, $y$ and $\delta$ from the time-independent measurements of coherent $(D^{0}\bar{D}^{0})\rightarrow(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ events. If the $CP$-violating phase of $D^{0}$-$\bar{D}^{0}$ mixing is significant in a scenario beyond the standard model, it can also be extracted from the decay modes under discussion.

The remaining part of this paper is organized as follows. Section 2 is devoted to the effects of $D^{0}$-$\bar{D}^{0}$ mixing and $CP$ violation in the time-dependent $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$ decays. The coherent $(D^{0}\bar{D}^{0})\rightarrow(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ decays on the $\psi(3770)$ and $\psi(4140)$ resonances are discussed in section 3, where we focus our interest on possible signals of $D^{0}$-$\bar{D}^{0}$ mixing and $CP$ violation in the time-independent events. An isospin analysis of the final state interactions in $D\rightarrow KK^{*}$ modes is done in section 4. Finally, we summarize our main results in section 5.

2   In the standard model $D^{0}\rightarrow K^{*\pm}K^{\mp}$ transitions can occur through both tree-level and loop-induced (penguin) quark diagrams. The former is essentially $CP$-conserving (proportional to $V_{cs}V^{*}_{us}$), while the latter is negligibly small (suppressed by $m^{2}_{q}/M^{2}_{W}$ for $q=d,s,b$) [3]. Hence the four amplitudes $A_{K^{*+}K^{-}}\equiv\langle K^{*+}K^{-}|{\cal H}|D^{0}\rangle$, $\bar{A}_{K^{*+}K^{-}}\equiv\langle K^{*+}K^{-}|{\cal H}|\bar{D}^{0}\rangle$, $A_{K^{*-}K^{+}}\equiv\langle K^{*-}K^{+}|{\cal H}|D^{0}\rangle$ and $\bar{A}_{K^{*-}K^{+}}\equiv\langle K^{*-}K^{+}|{\cal H}|\bar{D}^{0}\rangle$ have the relations $\bar{A}_{K^{*-}K^{+}}=A_{K^{*+}K^{-}}$ and $A_{K^{*-}K^{+}}=\bar{A}_{K^{*+}K^{-}}$ as a good approximation. We define

[TABLE]

where $\rho>0$ and $\delta$ is the strong phase difference. On the other hand, two neutral $D$-meson mass eigenstates can be written as

[TABLE]

where $p$ and $q$ satisfy the normalization condition $|p|^{2}+|q|^{2}=1$. The phase of $q/p$ is $\phi\equiv\arg(q/p)=\arg[(V^{*}_{cs}V_{us})/(V_{cs}V^{*}_{us})]\approx 0$ within the standard model [14], but it might be significant if a kind of new physics contributes to the box diagram of $D^{0}$-$\bar{D}^{0}$ mixing [3, 4, 5, 6, 12, 15]. Allowing for both $|q/p|\neq 1$ and $\phi\neq 0$, we may use the following rephasing-invariant quantities to express the decay rates of $D^{0}\rightarrow K^{*\pm}K^{\mp}$ and $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$:

[TABLE]

Since the naive factorization approximation yields $\rho\sim{\cal O}(1)$, it is quite natural to expect that $|\lambda_{K^{*+}K^{-}}|\approx|\bar{\lambda}_{K^{*-}K^{+}}|\sim{\cal O}(1)$ holds.

First, let us look at the time-dependent decay rates of $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$. Now that the $D^{0}$-$\bar{D}^{0}$ mixing parameters $x$ and $y$ are both small, we may just keep the terms of ${\cal O}(x)$ and ${\cal O}(y)$ in our calculations. Using the generic formulas given in Ref. [16] ‡‡‡Note that $y=-y_{D}$, where $y_{D}$ is the $D^{0}$-$\bar{D}^{0}$ mixing parameter defined in Ref. [16]., we explicitly obtain

[TABLE]

and

[TABLE]

where we have required $t\leq 1/\Gamma$ for the proper time $t$. Taking account of Eq. (4) and defining the effective $D^{0}$-$\bar{D}^{0}$ mixing parameters

[TABLE]

we simplify Eqs. (5) and (6) to

[TABLE]

and

[TABLE]

Once these four decay rates are measured, it will be possible to determine $\rho$ and constrain the magnitudes of both $D^{0}$-$\bar{D}^{0}$ mixing and $CP$ violation. Note that the deviation of $|p/q|$ (or $|q/p$) from unity, which can also be determined or constrained from other neutral $D$-meson decays, signifies $CP$ violation in $D^{0}$-$\bar{D}^{0}$ mixing. This effect is conveniently described by a small parameter $\Delta$ up to the correction of ${\cal O}(\Delta^{2})$; i.e., $p/q=1+\Delta$ and $q/p=1-\Delta$. Given $\phi\approx 0$ in the standard model, useful information on $y^{\prime}_{+}$ and $y^{\prime}_{-}$ is achievable from the time-dependent measurements of $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$ transitions. A clear difference between $y^{\prime}_{+}$ and $y^{\prime}_{-}$ will imply that both $x$ and $\delta$ are not very small. These points have also been observed in Ref. [15].

We remark that the $K^{*\pm}K^{\mp}$ events of neutral $D$-meson decays are important, since they can be complementary to the $K^{\pm}\pi^{\mp}$ and $K^{+}K^{-}$ (or $\pi^{+}\pi^{-}$) events for the experimental searches for both $D^{0}$-$\bar{D}^{0}$ mixing and $CP$ violation. A similar idea, which makes use of the $D^{*\pm}D^{\mp}$ events of neutral $B$-meson decays to extract the $CP$-violating phase $\beta$ and test the factorization hypothesis [18], has actually been adopted by the Belle [19] and BaBar [20] Collaborations in their experiments at the KEK and SLAC $B$ factories.

3   Now we turn to the possibility of measuring coherent $(D^{0}\bar{D}^{0})_{C}\rightarrow(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ decays on the $\psi(3770)$ resonance with $C=-1$ and (or) on the $\psi(4140)$ resonance with $C=+1$, where $C$ denotes the charge-conjugation parity of the $D^{0}$ and $\bar{D}^{0}$ pair. Both time-dependent and time-integrated rates of a general $(D^{0}\bar{D}^{0})_{C}\rightarrow f_{1}f_{2}$ decay mode, together with their approximate expressions up to the accuracy of ${\cal O}(x^{2})$ and ${\cal O}(y^{2})$, have been formulated in Ref. [16] without special assumptions. Here we focus our interest on the time-independent measurements of those $(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ events from coherent $(D^{0}\bar{D}^{0})_{C}$ decays at a high-luminosity $\tau$-charm factory (e.g., BEPC-II [13]). Let us define $\Gamma^{++}_{C}\equiv\Gamma(K^{*+}K^{-},K^{*+}K^{-})_{C}$, $\Gamma^{+-}_{C}\equiv\Gamma(K^{*+}K^{-},K^{*-}K^{+})_{C}$, $\Gamma^{-+}_{C}\equiv\Gamma(K^{*-}K^{+},K^{*+}K^{-})_{C}$ and $\Gamma^{--}_{C}\equiv\Gamma(K^{*-}K^{+},K^{*-}K^{+})_{C}$ for four joint decay rates. With the help of Ref. [16] §§§Note again that $y=-y_{D}$, where $y_{D}$ is the $D^{0}$-$\bar{D}^{0}$ mixing parameter defined in Ref. [16]., we explicitly have

[TABLE]

and $\Gamma^{-+}_{C}=\Gamma^{+-}_{C}$ with

[TABLE]

where $r\equiv(x^{2}+y^{2})/2$ is essentially the ratio of wrong-sign to right-sign events of semileptonic $D^{0}$ and $\bar{D}^{0}$ decays [16, 17]. When Eq. (4) is taken into account, Eqs. (10) and (11) can be simplified to

[TABLE]

and

[TABLE]

Note that the terms proportional to $r$ in $\Gamma^{\pm\pm}_{C}$ are only important when $C=-1$ is taken. As $K^{*+}K^{-}$ and $K^{*-}K^{+}$ are not the $CP$ eigenstates, both $\rho\neq 1$ and $\delta\neq 0$ are expected to hold. It is therefore reasonable to neglect the term proportional to $r$ in Eq. (13) even for the $C=-1$ case. We stress that these formulas will be very useful to analyze the experimental data on coherent $(D^{0}\bar{D}^{0})_{C}\rightarrow(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ decays at a $\tau$-charm factory.

For $C=-1$ on the $\psi(3770)$ resonance, we obtain

[TABLE]

where we have used the approximation $|p/q|\approx|q/p|\approx 1$ and neglected the term proportional to $r$ in $\Gamma^{+-}_{-}$. One can clearly see that these two ratios signify $D^{0}$-$\bar{D}^{0}$ mixing (i.e., $r\neq 0$). The difference between $\Gamma^{++}_{-}/\Gamma^{+-}_{-}$ and $\Gamma^{--}_{-}/\Gamma^{+-}_{-}$ measures the $CP$-violating effect in $D^{0}$-$\bar{D}^{0}$ mixing ($\Delta\neq 0$) and that from the interference between decay and mixing ($\phi\neq 0$):

[TABLE]

where the notations $|p/q|=1+\Delta$ and $|q/p|=1-\Delta$ have been taken into account. The smallness of $r$ (i.e., $r\sim 10^{-4}$), however, might more or less obstruct the observation of $\Gamma^{++}_{-}/\Gamma^{+-}_{-}$ and $\Gamma^{--}_{-}/\Gamma^{+-}_{-}$ at present. But we hope that the high-luminosity $\tau$-charm factory may finally realize the desired measurements in the near future.

For $C=+1$ on the $\psi(4140)$ resonance, one may simply neglect the terms proportional to $r$ in Eqs. (12) and (13). Up to small corrections of ${\cal O}(x^{\prime}_{\pm})$ and ${\cal O}(y^{\prime}_{\pm})$, the relationship

[TABLE]

holds approximately. Once the ratios $\Gamma^{++}_{+}/\Gamma^{+-}_{+}$ and $\Gamma^{--}_{+}/\Gamma^{+-}_{+}$ are measured, they will impose a strong constraint on $\rho$ and $\delta$. On the other hand, the difference between $\Gamma^{++}_{+}/\Gamma^{+-}_{+}$ and $\Gamma^{--}_{+}/\Gamma^{+-}_{+}$ is a clear signal of $CP$ violation:

[TABLE]

Comparing between $(\Gamma^{++}_{-}-\Gamma^{--}_{-})/\Gamma^{+-}_{-}$ and $(\Gamma^{++}_{+}-\Gamma^{--}_{+})/\Gamma^{+-}_{+}$, we find that the latter is less suppressed by the smallness of $x$ and $y$. Hence it seems more promising to measure $CP$ violation in the decays of correlated $D^{0}$ and $\bar{D}^{0}$ mesons into $(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ states on the $\psi(4140)$ resonance.

4   Finally let us make some comments on the final-state interactions in $D^{0}\rightarrow K^{*\pm}K^{\mp}$ transitions. A model-independent approach is to do the isospin analysis of $D^{0}\rightarrow K^{*+}K^{-}$, $D^{0}\rightarrow\bar{K}^{*0}K^{0}$ and $D^{+}\rightarrow K^{*+}\bar{K}^{0}$ decays or $D^{0}\rightarrow K^{*-}K^{+}$, $D^{0}\rightarrow K^{*0}\bar{K}^{0}$ and $D^{+}\rightarrow\bar{K}^{*0}K^{+}$ decays, in which each final state contains $I=0$ and (or) $I=1$ isospin configurations. For simplicity, we denote the amplitudes of $D^{0}\rightarrow K^{*+}K^{-}$, $D^{0}\rightarrow\bar{K}^{*0}K^{0}$ and $D^{+}\rightarrow K^{*+}\bar{K}^{0}$ as $A_{K^{*+}K^{-}}$, $A_{\bar{K}^{*0}K^{0}}$ and $A_{K^{*+}\bar{K}^{0}}$, respectively. They can be expressed in terms of two independent isospin amplitudes $A_{0}$ and $A_{1}$ as follows [16]:

[TABLE]

The branching ratios of these three decays are $B_{K^{*+}K^{-}}\propto|A_{K^{*+}K^{-}}|^{2}\tau_{0}$, $B_{\bar{K}^{*0}K^{0}}\propto|A_{\bar{K}^{*0}K^{0}}|^{2}\tau_{0}$ and $B_{K^{*+}\bar{K}^{0}}\propto|A_{K^{*+}\bar{K}^{0}}|^{2}\tau_{+}$, where $\tau_{0}=(410.1\pm 1.5)\times 10^{-15}~{}{\rm s}$ and $\tau_{+}=(1040\pm 7)\times 10^{-15}~{}{\rm s}$ are the lifetimes of $D^{0}$ and $D^{+}$ mesons [21], respectively. Defining $A_{0}/A_{1}=ze^{i\varphi}$, we find

[TABLE]

Then the isospin parameters $z$ and $\varphi$ can be determined:

[TABLE]

Of course, $\varphi\neq 0$ implies the existence of final-state interactions. One may follow a similar procedure to do the isospin analysis of $D^{0}\rightarrow K^{*-}K^{+}$, $D^{0}\rightarrow K^{*0}\bar{K}^{0}$ and $D^{+}\rightarrow\bar{K}^{*0}K^{+}$ decays. The amplitudes of these three transitions are essentially identical to those of $\bar{D}^{0}\rightarrow K^{*+}K^{-}$, $\bar{D}^{0}\rightarrow\bar{K}^{*0}K^{0}$ and $D^{-}\rightarrow K^{*0}K^{-}$ transitions, since their tree-level quark diagrams are $CP$-conserving and the penguin diagrams are negligibly small in the standard model. The corresponding isospin parameters $\bar{z}$ and $\bar{\varphi}$ can be extracted from the branching ratios $B_{K^{*-}K^{+}}$, $B_{K^{*0}\bar{K}^{0}}$ and $B_{\bar{K}^{*0}K^{+}}$ in the $CP$-conserving case. It is in general difficult to link the isospin phase differences $\varphi$ and $\bar{\varphi}$ to the strong phase difference $\delta$ defined in Eq. (2), unless some assumptions are made in a specific model of hadronic matrix elements. Nevertheless, it is reasonable to argue that significant $\varphi$ and $\bar{\varphi}$ must hint at significant $\delta$ for $K^{*\pm}K^{\mp}$ events.

For the purpose of illustration, we do a numerical analysis of the isospin parameters by using the present experimental data [21],

[TABLE]

and

[TABLE]

Since the magnitudes of $B_{\bar{K}^{*0}K^{0}}$ and $B_{K^{*0}\bar{K}^{0}}$ have not been fixed, our analysis can only provide some limited information on $(z,\varphi)$ and $(\bar{z},\bar{\varphi})$. The numerical results are shown in Fig. 1 and Fig. 2. Some comments are in order:

It is straightforward to see that the possibility of $\varphi=0$ and (or) $\bar{\varphi}=0$ is almost excluded by current experimental data. The most favorable values of $\varphi$ and $\bar{\varphi}$ are around $50^{\circ}$, implying the presence of significant final-state interactions. Indeed, $\varphi$ can be as large as $65^{\circ}$, and $\bar{\varphi}$ can be even larger than $80^{\circ}$. The strong phase difference $\delta$ is therefore expected to be significant in $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$ transitions. 2. 2.

The constraints on $B_{\bar{K}^{*0}K^{0}}/B_{K^{*+}\bar{K}^{0}}$ and $B_{K^{*0}\bar{K}^{0}}/B_{\bar{K}^{*0}K^{+}}$ allow us to extract the lower and (or) upper bounds of $B_{\bar{K}^{*0}K^{0}}$ and $B_{K^{*0}\bar{K}^{0}}$. We find $3.2\times 10^{-4}\leq B_{\bar{K}^{*0}K^{0}}\leq 1.6\times 10^{-3}$ and $B_{K^{*0}\bar{K}^{0}}<8\times 10^{-4}$. The former is interesting and can be tested in the upcoming experiments, but the latter is trivial. More accurate data will reduce the uncertainties in our isospin analysis. 3. 3.

The allowed ranges of $z$ and $\bar{z}$ do not have much overlap. In particular, $\bar{z}>0.8>z$ is roughly true. This observation implies that $D^{0}\rightarrow K^{*\pm}K^{\mp}$ and $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$ decays might involve quite different final-state interactions, from which significant $\delta$ is naturally anticipated.

It is worth mentioning that an isospin analysis of $D^{0}\rightarrow K^{+}K^{-}$, $D^{0}\rightarrow K^{0}\bar{K}^{0}$ and $D^{+}\rightarrow K^{+}\bar{K}^{0}$ decays [16], whose branching ratios have all been measured, also indicates the existence of strong final-state interactions. As $K^{+}K^{-}$ is a $CP$ eigenstate, however, the ratio of $\langle K^{+}K^{-}|{\cal H}|\bar{D}^{0}\rangle$ to $\langle K^{+}K^{-}|{\cal H}|D^{0}\rangle$ does not involve a significant strong phase difference in the absence of direct $CP$ violation [3].

5   In summary, we have investigated $D^{0}$-$\bar{D}^{0}$ mixing and $CP$ violation in $D^{0}\rightarrow K^{*\pm}K^{\mp}$ decays and their $CP$-conjugate processes, whose final states may have a significant strong phase difference. We have shown that both the $D^{0}$-$\bar{D}^{0}$ mixing parameters ($x$ and $y$) and the strong phase difference ($\delta$) can be determined or constrained from the time-dependent measurements of $D^{0}$ vs $\bar{D}^{0}\rightarrow K^{*\pm}K^{\mp}$ decays. For a high-luminosity $\tau$-charm factory running on the $\psi(3770)$ and $\psi(4140)$ resonances, we find that it is even possible to determine or constrain $x$, $y$ and $\delta$ from the time-independent measurements of coherent $(D^{0}\bar{D}^{0})\rightarrow(K^{*\pm}K^{\mp})(K^{*\pm}K^{\mp})$ events. If the $CP$-violating phase of $D^{0}$-$\bar{D}^{0}$ mixing is significant in a scenario beyond the standard model, it can also be extracted from the decay modes under discussion.

We strongly recommend the experimentalists to pay some special attention to the $K^{*\pm}K^{\mp}$ events of neutral $D$-meson decays, because they are complementary to the $K^{\pm}\pi^{\mp}$ and $K^{+}K^{-}$ (or $\pi^{+}\pi^{-}$) events for the study of both $D^{0}$-$\bar{D}^{0}$ mixing and $CP$ violation. We expect that these interesting channels and possible new physics in them can well be explored at BEPC-II and other charm-physics experiments in the near future.

One of us (Z.Z.X.) would like to thank Phil Chan and the National University of Singapore for warm hospitality, where this paper was written. He is also grateful to H.B. Li for some interesting discussions about the BaBar and Belle results. This work was supported in part by the National Natural Science Foundation of China.