---
abstract: |
    With heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll
    m_D\ll m_B$, we analyze the $B\to D^0\overline D^0$ and $B_s\to
    D^0\overline D^0$ decays, which occur purely via annihilation type diagrams. As a roughly estimation, we calculate their branching ratios and CP asymmetries in Perturbative QCD approach. The branching ratio of $B\to D^0\overline D^0$ is about $3.8\times10^{-5}$ that is just below the latest experimental upper limit. The branching ratio of $B_s\to D^0\overline D^0$ is about $6.8\times10^{-4}$, which could be measured in LHC-b. From the calculation, it could be found that this branching ratio is not sensitive to the weak phase angle $\gamma$. In these two decay modes, there exist CP asymmetries because of interference between weak and strong interaction. However, these asymmetries are too small to be measured easily.
author:
- |
    Ying Li[^1]  and Juan Hua\
    [*Physics Department, Yantai University, Yantai 264005, China*]{}
title: 'Study of Pure Annihilation Decays $B_{d,s} \to D^{0} \overline D^{0}$ '
---

Introduction {#sc:intro}
============

In the Standard Model (SM), CP-violation (CPV) arises from a complex phase in the Cabibbo-Kobayashi- Maskawa (CKM) quark mixing matrix, and the angles of unitary triangle are defined as [@Yao:2006px]: $$\begin{aligned}
\beta=\arg \Bigl[-\frac{V_{cb}^*V_{cd}}{V_{tb}^*V_{td}}\Bigl],~~~~
\alpha=\arg \Bigl[-\frac{V_{tb}^*V_{td}}{V_{ub}^*V_{ud}}\Bigl],~~~~
\gamma=\arg
\Bigl[-\frac{V_{ub}^*V_{ud}}{V_{cb}^*V_{cd}}\Bigl].\label{ckmangle}\end{aligned}$$ In order to test SM and search for new physics, many measurements of CP-violation observables can be used to constrain these above angles. It is well known that we measure $\beta$ precisely using the golden decay mode $B\to J/\psi K_s$; the angle $\alpha$ can be determined with decay $B\to\pi\pi$ and $\gamma$ could be measured precisely in Large Hadron Collider (LHC) with decay mode $B_s\to
D_sK$.

![The quark level Feynman diagrams for $B_d \to D^{0} \bar
D^{0}$ process []{data-label="fig0"}](fig1.eps)

Besides the above channels mentioned, many other channels are used to cross check the measurements. Among these decays, $B\to DD$ decay is considered to test the $\beta$ measurement. For $B\to DD$ decay, the analysis based on $SU(3)$ symmetry [@Savage:1989ub], iso-spin symmetry [@Xing:1999yx] and factorization approach [@Xing:1998ca] have been done in last several years. However, the the calculation of decay $B^0 \to D^0\overline D^0$ has difficulties. It is a pure-annihilation diagram decay, also named W-exchange diagram decay, which is power suppressed in factorization language. The quark diagrams of this decay are shown in Figure \[fig0\]. Theoretically, QCD factorization approach (QCDF) [@Beneke:1999br] and soft collinear effective theory (SCET)[@Bauer:2001yt] can not deal decays with two heavy charmed mesons effectively. In Ref.[@Keum:2003js; @Lu:2003xc], perturbative QCD (PQCD) has been exploited to $B$ meson decays with one charm meson in the final states and the results agree with experimental data well. Specially, the pure annihilation type B decays with charmed meson were studied in Ref.[@Lu:2003xc].

In the standard model picture, the $W$ boson exchange causes $\bar{b}d \to \bar{c}c $, and the $\bar{u}u$ quarks are produced from a gluon. This gluon attaches to any one of the quarks participating in the $W$ boson exchange. In decay $B \to D^{0}
\overline D^{0}$, the momentum of the final state $D$ meson is $\frac{1}{2} m_B (1-2 r^2)$, with $r=m_D/m_B$. If we consider heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll m_D\ll
m_B$, the $D$ meson momentum is nearly $m_B/2$. According to the distribution amplitude used in Ref.[@Keum:2003js], the light quark in $D$ meson carrying nearly $40\%$ of the $D$ meson momentum. So, this light quark is still a collinear quark with 1 $\mathrm{GeV}$ energy, like that in $B\to DM$ [@Keum:2003js; @Lu:2003xc], $B\to K(\pi)\pi$ [@kls; @luy] decays. The gluon could be viewed as a hard gluon approximatively, so we can treat the process perturbatively where the four-quark operator exchanges a hard gluon with $u \bar u$ quark pair. Of course, we are able to calculate the diagrams if charm quark and up quark exchange. As a roughly estimation, we give the branching ratio and CP-violation of $B_{d,s} \to D^{0} \overline D^{0}$.

In this article, the analytic formulas for the decay amplitudes will be shown in the next section. In section \[sc:result\], we give the numerical results and summarize this article in section \[sc:summary\].

Analytic formulas {#sc:analy}
=================

For simplicity, we set $B$ meson at rest in our calculation. In light-cone coordinates, the momentum of $B$, $D^0$ and$\overline
D^0$ are: $$\begin{aligned}
P_B=\frac{M_B}{\sqrt{2}}(1,1,\vec{0});
P_2=\frac{M_B}{\sqrt{2}}(1-r^2,r^2,\vec{0});
P_3=\frac{M_B}{\sqrt{2}}(r^2,1-r^2,\vec{0}).\end{aligned}$$ we define the light (anti-)quark momenta in $B$, $D^0$ and $\overline D^0$ mesons as $k_1$, $k_2$, and $k_3$ as: $$k_1 =
(x_1P_1^+,0,{\bf k}_{1T}),\ \  k_2 = (x_2 P_2^+,0,{\bf k}_{2T}),\ \
 k_3 = (0, x_3 P_3^-,{\bf k}_{3T}). \label{eq:momentun2}$$

In PQCD, we factorize the decay amplitude into soft($\Phi$), hard($H$), and harder ($C$) dynamics characterized by different scales, [@kls; @luy] $$\begin{gathered}
\mathcal{A}\sim
 \int\!\! d x_1 d x_2 d x_3 b_1 d b_1 b_2 d b_2 b_3
d b_3  \mathrm{Tr} \Bigl[ C(t) \Phi_B(x_1,b_1) \Phi_{D}(x_2,b_2)
\Phi_D(x_3, b_3) H(x_i, b_i,t) S_t(x_i)\, e^{-S(t)} \Bigr].
\label{eq:convolution2}\end{gathered}$$ In above equation, $b_i$ is the conjugate space coordinate of the transverse momentum ${\bf k}_{iT}$, and $t$ is the largest energy scale. $C$ is Wilson coefficient, and $\Phi$ is the wave function. The last term, $e^{-S(t)}$, contains two kinds of contributions. One is due to the resummation of the large double logarithms from renormalization of ultra-violet divergence $\ln tb$, the other is from resummation of double logarithm $\ln^2 b$ from the overlap of collinear and soft gluon corrections, which is called Sudakov form factor. The hard part $H$ can be calculated perturbatively, and it is channel dependent. More explanation of above formula and review about PQCD can be found in many reference, such as [@kls; @luy; @Ali:2007ff].

As a heavy meson, the $B$ meson wave function is not well defined, neither is $D$ meson. In heavy quark limit, we take them as: $$\Phi_{B}(x,b) = \frac{i}{\sqrt{6}}
\left[ \not \! P  + M_B  \right] \gamma_5\phi_B(x,b),$$ $$\Phi_{D}(x,b) = \frac{i}{\sqrt{6}}\gamma_5
\left[ \not \! P  +  M_D \right] \phi_D(x,b).$$ The Lorentz structure of two mesons are different because the $B$ meson is initials state and $D$ meson is final state.

The effective Hamiltonian $\bar b\to \bar q(q=d,s)$ is given by [@Buchalla:1996vs]: $$\begin{gathered}
 \mathcal{H}_\mathrm{eff} =
 \frac{G_F}{\sqrt{2}}\Bigl\{ V_{cq}V_{cb}^* \Bigl[
C_1(\mu) O_1^c(\mu) + C_2(\mu) O_2^c(\mu) \Bigl]+ V_{uq}V_{ub}^*
\Bigl[ C_1(\mu) O_1^u(\mu) + C_2(\mu) O_2^u(\mu) \Bigl]\\
-V_{tb}^*V_{tq}\sum_{i=3}^{10}C_i(\mu) O_i(\mu)\Bigl\},\label{hami}\end{gathered}$$ where $C_{i}(\mu)(i=1,\cdots,10)$ are Wilson coefficients at the renormalization scale $\mu$ and the four quark operators $O_{i}(i=1,\cdots,10)$ are $$\begin{array}{ll}
  O_1^c = (\bar{b}_ic_j)_{V-A}(\bar{c}_jq_i)_{V-A},  &
  O_2^c = (\bar{b}_ic_i)_{V-A} (\bar{c}_jq_j)_{V-A},  \\
  O_1^u = (\bar{b}_iu_j)_{V-A}(\bar{u}_jq_i)_{V-A},  &
  O_2^u = (\bar{b}_iu_i)_{V-A} (\bar{u}_jq_j)_{V-A},  \\
  O_3 = (\bar{b}_iq_i)_{V-A}\sum_{q} (\bar{q}_jq_j)_{V-A},  &
  O_4 = (\bar{b}_iq_j)_{V-A}\sum_{q} (\bar{q}_jq_i)_{V-A}, \\
  O_5 = (\bar{b}_iq_i)_{V-A}\sum_{q} (\bar{q}_jq_j)_{V+A},  &
  O_6 = (\bar{b}_iq_j)_{V-A} \sum_{q} (\bar{q}_jq_i)_{V+A}, \\
  O_7 = \frac{3}{2}(\bar{b}_iq_i)_{V-A} \sum_{q}
   e_q(\bar{q}_jq_j)_{V+A},   &
   O_8 = \frac{3}{2}(\bar{b}_iq_j)_{V-A}\sum_{q} e_q
  (\bar{q}_jq_i)_{V+A}, \\
  O_9 = \frac{3}{2}(\bar{b}_iq_i)_{V-A}\sum_{q}
  e_q(\bar{q}_jq_j)_{V-A}, &
   O_{10} = \frac{3}{2}(\bar{b}_iq_j)_{V-A}\sum_{q}
 e_q(\bar{q}_jq_i)_{V-A}. \label{eq:effectiv}
 \end{array}$$ Here $i$ and $j$ are $SU(3)$ color indices; in $O_{3,...,10}$ the sum over $q$ runs over the quark fields that are active at the scale $\mu=O(m_{b})$, i.e., $q\in \{u,d,s,c,b\}$. For Wilson coefficients, we will also use the leading logarithm summation for QCD corrections, although the next-to -leading order calculation already exists [@Buchalla:1996vs]. This is the consistent way to cancel the explicit $\mu$ dependence in the theoretical formulae.

![The leading order Feynman diagrams for $B_d \to D^{0}
\overline D^{0}$ process in PQCD approach[]{data-label="fig1"}](fig2.eps)

According to the effective Hamiltonian in eq.(\[hami\], \[eq:effectiv\]), the lowest order diagrams of $B \to D^0\overline
D^0$ are drawn in Fig. \[fig1\]. We first calculate the usual factorizable diagrams (a), (b), (c) and (d). For the $(V-A)(V-A)$ operators, their contributions of (a) and (c) are always canceled by diagrams (b) and (d) respectively because of current conservation. For the $(V-A)(V+A)$ operators, these diagrams can not give contribution, either. That’s to say, factorizable diagrams have no contribution. For non-factorizable diagrams (e), (f), (g) and (h), we find the hard part for $(V-A)(V-A)$ operators are same to $(V-A)(V+A)$ operators. We group the contribution of diagrams (e) and (f), denoted by $M_a$, as follows: $$\begin{gathered}
M_{a}[C_i]  =  \frac{64 \pi C_FM_B^2}{\sqrt{2N_C}}  \int_0^1 \!\!
dx_1 dx_2 dx_3
 \int_0^\infty \!\! b_1 db_1\, b_2 db_2\
\phi_B(x_1,b_1) \phi_{D}(x_2,b_2)\phi_{D}(x_3,b_2) \\
\times \Bigl\{ \Bigl[x_1+x_2+(2x_3-x_2)r^2\Bigl]
C_i(t_{a}^1)E(t_{a}^1) h_a^{(1)}(x_1,
x_2,x_3,b_1,b_2) \\
+\Bigl[-x_3+(2x_1-2x_2+x_3)r^2\Bigl] C_i(t_{a}^2)E(t_{a}^2)
h_a^{(2)}(x_1, x_2,x_3,b_1,b_2) \Bigr\}, \label{eq:Ma}\end{gathered}$$ where $C_F = 4/3$ is the group factor of $\mathrm{SU}(3)_c$ gauge group, and $C_i$ is Wilson coefficient. The function $E_m$ is defined as $$E(t) = \alpha_s(t)\, e^{-S_B(t)-S_D(t)-S_D(t)},$$ and $S_B$, $S_D$ result from Sudakov factor and single logarithms due to the renormalization of ultra-violet divergence. The functions $h_a$ is the Fourier transformation of virtual quark and gluon propagators. It is defined by $$\begin{aligned}
 &h^{(j)}_a(x_1,x_2,x_3,b_1,b_2) = \nonumber \\
 & \biggl\{ \frac{\pi i}{2}
 \mathrm{H}_0^{(1)}(M_B\sqrt{x_2x_3(1-2r^2)}\, b_1)
  \mathrm{J}_0(M_B\sqrt{x_2x_3(1-2r^2)}\, b_2) \theta(b_1-b_2)
 \nonumber \\
 & \qquad\qquad\qquad\qquad + (b_1 \leftrightarrow b_2) \biggr\}
  \times\left(
 \begin{matrix}
  \mathrm{K}_0(M_B F_{a(j)} b_1), & \text{for}\quad F^2_{a(j)}>0 \\
  \frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{|F^2_{a(j)}|}\ b_1), &
  \text{for}\quad F^2_{a(j)}<0
 \end{matrix}\right),
 \label{eq:propagator1}
 \end{aligned}$$ with: $$\begin{aligned}
  F^2_{a(1)} &=&-x_1-x_2-x_3+x_1x_3+x_2x_3+(x_2+x_3-x_1x_3-2x_2x_3)r^2;\\
  F^2_{a(2)}&=&x_2x_3-x_1x_3+(x_1x_3-2x_2x_3)r^2.\end{aligned}$$ In above equation, $\mathrm{H}_0^{(1)}(z) = \mathrm{J}_0(z) + i\,
 \mathrm{Y}_0(z)$. In order to reduce the large logarithmic radiative corrections, the hard scale $t$ in the amplitudes is selected as the largest energy scale in the hard part: $$\begin{aligned}
  t_{a}^j &=& \mathrm{max}(M_B \sqrt{|F^2_{a(j)}|},
 M_B \sqrt{(1-2r^2)x_2x_3 }, 1/b_1,1/b_2).
 \end{aligned}$$ Analogically, we can get the $M_{b}$, which comes from the contribution of diagrams (g) and (h): $$\begin{gathered}
M_{b}[C_i]=  \frac{64 \pi C_FM_B^2}{\sqrt{2N_C}}  \int_0^1 \!\! dx_1
dx_2 dx_3
 \int_0^\infty \!\! b_1 db_1\, b_2 db_2\
\phi_B(x_1,b_1) \phi_{D}(x_2,b_2)\phi_{D}(x_3,b_2) \\
\times \Bigl\{ \Bigl[1-x_3+(2+2x_1-2x_2+x_3)r^2\Bigl]C_i(t_{b}^1)
E(t_{b}^1) h_b^{(1)}(x_1,
x_2,x_3,b_1,b_2) \\
+\Bigl[x_1+x_2-1+(-2-x_2+2x_3)r^2\Bigl]C_i(t_{b}^2) E(t_{b}^2)
h_b^{(2)}(x_1, x_2,x_3,b_1,b_2) \Bigr\}, \label{eq:Mb}\end{gathered}$$ and the functions are defined as: $$\begin{aligned}
 &h^{(j)}_b(x_1,x_2,x_3,b_1,b_2) = \nonumber \\
 & \biggl\{ \frac{\pi i}{2}
 \mathrm{H}_0^{(1)}(M_B\sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2}\,
 b_1) \nonumber\\
&~~~~~~~~
\times\mathrm{J}_0(M_B\sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2}\,
b_2) \theta(b_1-b_2)
 \nonumber \\
 & \qquad\qquad\qquad\qquad + (b_1 \leftrightarrow b_2) \biggr\}
  \times\left(
 \begin{matrix}
  \mathrm{K}_0(M_B F_{b(j)} b_1), & \text{for}\quad F^2_{b(j)}>0 \\
  \frac{\pi i}{2} \mathrm{H}_0^{(1)}(M_B\sqrt{|F^2_{b(j)}|}\ b_1), &
  \text{for}\quad F^2_{b(j)}<0
 \end{matrix}\right);
 \label{eq:propagator2}
 \end{aligned}$$ $$\begin{aligned}
  F^2_{b(1)}&=&-1-x_1x_3+x_2x_3+(x_1x_3-2x_2x_3)r^2,\\
  F^2_{b(2)}&=&1-x_1-x_2-x_3+x_1x_3+x_2x_3+(x_2+x_3-x_1x_3-2x_2x_3)r^2,
 \end{aligned}$$ $$\begin{aligned}
  t_{b}^j &=& \mathrm{max}(M_B \sqrt{|F^2_{b(j)}|},
 M_B \sqrt{1-x_2-x_3+x_2x_3+(x_2+x_3-2x_2x_3)r^2 }, 1/b_1,1/b_2).
  \end{aligned}$$ So, the decay amplitude of decay $B_d \to D^{0} \overline D^{0}$ can be read as: $$\begin{aligned}
\mathcal{A}_1&=&V_{cb}^*V_{cd}M_{a}[C_2]-V_{tb}^*V_{td}M_{a}[C_5+C_7]+V_{ub}^*V_{ud}M_{b}[C_2]
-V_{tb}^*V_{td}M_{b}[C_5+C_7]\nonumber\\
           &=&V_{cb}^*V_{cd}T_1
           -V_{tb}^*V_{td}P_1\nonumber\\
           &=&V_{tb}^*V_{td}P_1(1+z_1e^{i(\beta+\delta_1)}),\label{bdd1}\end{aligned}$$ where $\beta$ is weak phase angle defined in Eq.(\[ckmangle\]), and $\delta_1$ is the strong phase, which plays an important role in studying CP-violation. In above calculation, we denote that $$\begin{aligned}
T_1&=&M_{a}[C_2]-M_{b}[C_2], \nonumber \\
P_1&=&M_{a}[C_5+C_7]+M_{b}[C_5+C_7]+M_{b}[C_2],\end{aligned}$$ and $$\begin{aligned}
z_1=\left|\frac{V_{cb}^*V_{cd}}{V_{tb}^*V_{td}}
\right|\left|\frac{T_1}{P_1}\right|,\end{aligned}$$ which describes the ratio between tree diagram and penguin diagram. The corresponding charge conjugate decay is $$\begin{aligned}
\label{bdd2}
\overline{\mathcal{A}_1}=V_{tb}V_{td}^*P_1(1+z_1e^{i(-\beta+\delta_1)}).\end{aligned}$$ Therefore, the averaged decay width $\Gamma$ for $B^0 \to  D^0
\overline D^0$ decay is then given by $$\Gamma(B^0 \to  D^0
\overline D^0) = \frac{G_F^2 M_B^3}{128\pi}
 (1-2r^2)|V_{tb}^*V_{td}P_1|^2
\bigl|1+z_1^2+2z_1\cos \beta \cos\delta_1|. \label{eq:neut_width1}$$ &gt;From this equation, we know that the averaged branching ratio is a function of CKM angle $\beta$, if $z_1\neq 0$. Derived from Eq.(\[bdd1\]) and Eq.(\[bdd2\]), the direct CP-violation can be formulated as: $$\begin{aligned}
\label{cp1}
A_{CP}^{dir}(B\to D^{0} \overline D^{0})=\frac{|A_{B_d\to
D^0\overline D^0}|^2-|A_{\overline B_d\to \overline
D^0D^0}|^2}{|A_{B_d\to D^0\overline D^0}|^2+|A_{\overline B_d\to
\overline
D^0D^0}|^2}=\frac{-2z_1\sin\beta\sin\delta_1}{1+z_1^2+2z_1\cos \beta
\cos\delta_1}.
 \end{aligned}$$

For $ B^0_s \to  D^0 \overline D^0$ and its conjugate decay, we write the decay amplitudes and rearrange them as: $$\begin{aligned}
\mathcal{A}_2&=&V_{cb}^*V_{cs}M_{a}[C_2]-V_{tb}^*V_{ts}M_{a}[C_5+C_7]+V_{ub}^*V_{us}M_{b}[C_2]
-V_{tb}^*V_{ts}M_{b}[C_5+C_7]\nonumber\\
            &=&V_{ub}^*V_{us}M_{b}[C_2]-V_{tb}^*V_{ts}\Bigl\{M_{a}[C_5+C_7]
            +M_{b}[C_5+C_7]-\frac{V_{cb}^*V_{cs}}{V_{tb}^*V_{ts}}M_{a}[C_2]\Bigl\}\nonumber\\
            &=&V_{ub}^*V_{us}T_2-V_{tb}^*V_{ts}P_2\nonumber\\
            &=&V_{ub}^*V_{us}T_2\Bigl[1+z_2e^{i(-\gamma+\delta_2)}\Bigl],\\
\overline{\mathcal{A}_2}&=&V_{ub}V_{us}^*T_2\Bigl[1+z_2e^{i(\gamma+\delta_2)}\Bigl],\label{bdd3}\end{aligned}$$ where $T_2$, $P_2$ and $z_2$ are defined as: $$\begin{aligned}
T_2&=&M_{b}[C_2],\nonumber\\
 P_2&=&M_{a}[C_5+C_7]
            +M_{b}[C_5+C_7]-\frac{V_{cb}^*V_{cd}}{V_{tb}^*V_{ts}}M_{a}[C_2],\nonumber\\
z_2&=&\left|\frac{V_{tb}^*V_{ts}}{V_{ub}^*V_{us}}
\right|\left|\frac{T_2}{P_2}\right|.\end{aligned}$$ So, the averaged decay width and direct CP violation can be formulated as: $$\begin{aligned}
\label{eq:neut_width2}
 \Gamma(B_s \to  D^0\overline D^0) &=& \frac{G_F^2 M_B^3}{128\pi} (1-2r^2) \bigl|V_{ub}V_{us}^*T_2\bigr|^2
 (1+z_2^2+2z_2\cos\delta_2\cos\gamma),\\
A_{CP}^{dir}(B_s \to D^0\overline D^0)&=&\frac{|A_{B_s\to
D^0\overline D^0}|^2-|A_{\overline B_s\to \overline
D^0D^0}|^2}{|A_{B_s\to D^0\overline D^0}|^2+|A_{\overline B_s\to
\overline
D^0D^0}|^2}=\frac{2z_2\sin\gamma\sin\delta_2}{1+z_2^2+2z_2\cos
\gamma \cos\delta_2}.\end{aligned}$$ In our calculation, we set $m_c\approx m_D$, just because $m_D -m_c
\approx \Lambda_{QCD}$ and $\frac{\Lambda_{QCD}}{m_B} \rightarrow 0$ in the heavy quark limit.

Numerical Results {#sc:result}
=================

For $B$ meson, the distribution amplitude is well determined by charmless $B$ decays [@kls; @luy], which is chosen as $$\phi_B(x,b) = N_B x^2(1-x)^2 \exp \left[ -\frac{M_B^2\ x^2}{2
\omega_b^2} -\frac{1}{2} (\omega_b b)^2 \right],\label{waveb}$$ with parameters $\omega_b=0.4\mbox{ GeV}$, and $N_B=91.745\mbox{
GeV}$ which is the normalization constant using $f_B=190 \mbox{
MeV}$. For $B_s$ meson, we use the same wave function according to SU(3) symmetry, where $\omega_b=0.4\mbox{ GeV}$, $N_{B_s}=119.4
\mbox{ GeV}$ and $f_{B_s}=230 \mbox{ MeV}$.

Since the $c$ quark is much heavier than the $u$ quark, the $c$ quark shares more momentum, and this function should be asymmetric with respect to $x = 1/2$. The asymmetry is parameterized by $a_D$. Similar to the $b$-dependence on the wave function of $B$ meson, for controlling the size of charmed mesons, we also introduce the intrinsic $b$-dependence on those of charmed mesons. Hence, we use the wave function of $D$ meson as [@Chen:2003px] $$\phi_{D}(x,b) = \frac{3}{\sqrt{2 N_c}} f_{D} x(1-x)\Bigl[ 1 +
a_{D} (1 -2x)\Bigl]\exp \left[-\frac{1}{2} (\omega_D b)^2
\right].\label{waved}$$ We use $a_{D}=0.7$ and $\omega_{D}=0.4$ in above function. Other parameters, such as meson mass, decay constants, the CKM matrix elements and the lifetime of $B$ meson are list [@Yao:2006px; @Follana:2007uv]: $$\begin{gathered}
M_B = 5.28 \mbox{ GeV},\  M_{B_S} = 5.36 \mbox{ GeV},\  M_{D} = 1.87
\mbox{ GeV},\ f_{D} = 210
\mbox{ MeV}, \nonumber \\
|V_{ud}|=0.974, \  |V_{ub}|= 4.3 \times 10^{-3},\   |V_{cd}|=0.23,
\   |V_{cb}|= 41.6\times 10^{-3} \nonumber\\
|V_{td}|= 7.4\times 10^{-3}, \  |V_{tb}|=1.0,\
|V_{us}|=0.226, \  |V_{cs}|=0.957,\nonumber \\
|V_{ts}|=41.6\times 10^{-3}, \quad\tau_{B_d^0}=1.54\times 10^{-12}\mbox{ s,}\quad
 \tau_{B_s^0}=1.46\times 10^{-12}\mbox{ s}.\label{parameterZ}\end{gathered}$$

                 $B_d\to D^0\overline D^0$   $B_s\to D^0\overline D^0$
  ------------- --------------------------- ---------------------------
    $T(e)+T(f)$        $68+17 \, i$                $66 +27 \, i$
    $P(e)+P(f)$      $ 0.80+ 0.23\, i$           $0.77 +3.68\, i$
    $T(g)+T(h)$      $ 9.81- 2.99\, i$            $14.0-0.6\, i$
    $P(g)+P(h)$      $ 0.08- 0.02\, i$           $-0.01 +0.01\, i$

  : Amplitudes ($10^{-3}$ GeV) of $B_d\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$.[]{data-label="tb:amplitudes"}

![The branching ratio of $B_s\to D^0\overline D^0$ changes with CKM angle $\gamma$.[]{data-label="fig3"}](fig3.eps)

With these parameters fixed, we calculate the decay amplitudes of the $B^0\to D^0\overline D^0$ and $B_s\to D^0\overline D^0$ decays in Table \[tb:amplitudes\]. From the table, we notice that the main contribution comes from the tree diagram (e) and (f). And our predictions for the branching ratio of each mode corresponding to $\beta=23^\circ$ and $\gamma=63^\circ$ are listed, $$\begin{aligned}
  BR(B_d\to D^0\overline D^0) &=& 2.3\times 10^{-5}; \nonumber\\
  BR(B_s\to D^0\overline D^0) &=& 6.8\times 10^{-4}.\end{aligned}$$ In Fig. \[fig3\], we plot the branching ratio of $B_s\to
D^0\overline D^0$ with different $\gamma$. In this figure, we find the branching ratio is not sensitive to CKM angle $\gamma$. For the experimental side, there are only upper limits given at $90\%$ confidence level for decay $B_d\to D^0\overline D^0$, $$\begin{aligned}
  BR(B_d\to D^0\overline D^0) &< &6.0\times 10^{-5}; ~~~~~~BarBar\cite{Aubert:2006ia}\nonumber\\
  BR(B_d\to D^0\overline D^0) &< &4.2\times 10^{-5}. ~~~~~~Belle\cite{:2007sk}\end{aligned}$$ Obviously, our result is consistent with the data. For $B_d\to
D^0\overline D^0$ decay mode, $z_1$ is about 6.5, and the strong phase $\delta_1$ is $34^\circ$, so $A_{CP}^{dir}$ is about $-6\%$ with the definition in Eq.(\[cp1\]). As decay mode $B_s\to
D^0\overline D^0$ is concerned, $z_2$ is about $205$ and $\delta_2=155^\circ$, and the relation between direct CP violation and $\gamma$ is shown in Fig.\[fig4\]. From the figure, we read the CP asymmetry is about $0.4\%$, which is rather tiny. It is necessary to state that the $z_1$ and $z_2$ are not the true ratio between tree contribution and penguin, because mathematical technics are used in Eq. (\[bdd1\]) and (\[bdd3\]).

![The direct CP-violation of $B_s\to D^0\overline D^0$ changes with CKM angle $\gamma$.[]{data-label="fig4"}](fig4.eps)

In addition to the perturbative annihilation contributions, there is also a hadronic picture for the $B_d\to D^0\overline D^0$, named soft final states interaction[@Cheng:2004ru]. The $B$ meson decays into $D^+$ and $D^-$, the secondary particles then exchanging a $\rho$ meson, then scatter into $D^0\overline D^0$ through final state interaction afterwards. For $B_s$ decay, the $B_s$ meson decays into $D_s^+$ and $D^+$ then scatters into $D^0\overline D^0$ by exchanging a Kaon. But this picture cannot be calculated accurately because of lack of many effective vertexes, and we will ignore this contribution here, though it may be important [@Cheng:2004ru].

  ---------------------------- ------------------------------- ------------------------------- ----------------------------------------- -----------------------------------------
                                $BR(B_d\to D^0\overline D^0)$   $BR(B_s\to D^0\overline D^0)$   $A_{CP}^{dir}(B_d\to D^0\overline D^0)$   $A_{CP}^{dir}(B_s\to D^0\overline D^0)$
                                     $(\times 10^{-5})$              $(\times 10^{-4})$                         $(\%)$                                    $(\%)$
    $\omega_b(B\setminus B_s)$                                                                                                           
           $0.35\setminus0.45$               4.3                             7.8                                 -7.2                                       0.4
           $0.40\setminus0.50$               3.8                             6.8                                 -5.3                                       0.4
           $0.45\setminus0.55$               3.2                             5.9                                 -5.8                                       0.4
                    $\omega_D$                                                                                                           
                        $0.35$               5.0                             9.7                                 -4.2                                       0.3
                        $0.40$               3.8                             6.8                                 -5.3                                       0.4
                        $0.45$               2.2                             4.2                                 -7.8                                       0.5
                         $a_D$                                                                                                           
                         $0.6$               3.2                             5.9                                 -6.9                                       0.4
                         $0.7$               3.8                             6.8                                 -5.3                                       0.4
                         $0.8$               4.3                             7.8                                 -6.1                                       0.4
  ---------------------------- ------------------------------- ------------------------------- ----------------------------------------- -----------------------------------------

  : The sensitivity of the decay branching ratios and CP asymmetries to change of ¥ø$\omega_b$, $\omega_D$ and $a_D$[]{data-label="table"}

There are many uncertainties in our calculation such as higher order corrections, the parameters listed in Eq.(\[parameterZ\]) and the distribution amplitudes of heavy mesons. We will not discuss uncertainty taken by high order correction as we only roughly estimate the branching ratios and CP asymmetries, though high order corrections have been done for some special channels [@Li:2005kt; @Li:2006jv] and showed $15-20\%$ uncertainty. The parameters in Eq.(\[parameterZ\]), fixed by experiments, are proportional to the amplitudes, so we will not analyze this kind uncertainty either. In our calculation, we find that the results are sensitive to the distribution amplitudes, especially to that of $D$ meson. Since the heavy $D$ wave function is less constrained, we set $a_{D}\in (0.6-0.8)\mbox{ GeV}$ and $\omega_{D}\in(0.35-0.45)\mbox{
GeV}$ to exploit the uncertainty. Table \[table\] shows the sensitivity of the branching ratios to change of $\omega_b$, $\omega_D$ and $a_{D}$. It is found that uncertainty of the predictions on PQCD is mainly due to $\omega_D$, which describes the behavior in end-point region of $D$ meson, however it is very hard to be determined. Considering the experimental upper limit, our results favor large $\omega_b$, large $\omega_D$ and small $a_D$.

At last, we give the prediction of branching ratios with err bar as follows: $$\begin{aligned}
  BR(B_d\to D^0\overline D^0) &=&(3.8^{+0.5+1.2+0.5}_{-0.6-1.6-0.6})\times 10^{-5}
  \left(\frac{f_B\cdot f_D\cdot f_D}{190\mathrm{ GeV}\cdot 210\mathrm{ GeV}\cdot 210\mathrm{ GeV}}\right)^2;  \nonumber\\
  BR(B_s\to D^0\overline D^0) &=&(6.8^{+1.0+2.9+1.0}_{-0.9-2.6-0.9})\times 10^{-4}
  \left(\frac{f_{B_s}\cdot f_D\cdot f_D}{230\mathrm{ GeV}\cdot 210\mathrm{ GeV}\cdot 210\mathrm{ GeV}}\right)^2.\end{aligned}$$ We believe that the $B_d\to D^0\overline D^0$ will be measured soon because this ratio is just below the upper limit, and $B_d\to
D^0\overline D^0$ will be measured in LHC-b in next year as a channel to cross check the $\gamma$ measurements.

Summary {#sc:summary}
=======

With heavy quark limit and hierarchy approximation $\lambda_{QCD}\ll
m_D\ll m_B$, we analyze the $B\to D^0\overline D^0$ and $B_s\to
D^0\overline D^0$ decays, which occur purely via annihilation type diagrams. As a roughly estimation, we calculate the branching ratios and CP asymmetries in PQCD approach. The branching ratios are still sizable. The branching ratio of $B\to D^0\overline D^0$ is about $3.8\times10^{-5}$, which is just below the experimental upper limited result[@Aubert:2006ia; @:2007sk], and we think that it will be measured in near future. For $B_s\to D^0\overline D^0$, the branching ratio is about $6.8\times10^{-4}$, which could be measured in LHC-b. From the calculation, it is found that this branching ratio is not sensitive to angle $\gamma$. In these two decays, there exist CP asymmetries because of interference between weak and strong interaction, though they are very small.

Acknowledgements {#acknowledgements .unnumbered}
================

Y.Li thanks Institute of High Energy Physics for their hospitality during visit where part of this work was done. This work is partly supported by Foundation of Yantai University under Grant No.WL07B19. We would like to acknowledge C.D. Lü and W. Wang for valuable discussions.

[99]{} W. M. Yao [*et al.*]{} \[Particle Data Group\], J. Phys. G [**33**]{}, 1 (2006). M. J. Savage and M. B. Wise, Phys. Rev.  D [**39**]{}, 3346 (1989) \[Erratum-ibid.  D [**40**]{}, 3127 (1989)\]. Z. Z. Xing, Phys. Rev.  D [**61**]{}, 014010 (2000) \[arXiv:hep-ph/9907455\].

Z. Z. Xing, Phys. Lett.  B [**443**]{}, 365 (1998) \[arXiv:hep-ph/9809496\].

M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Phys. Rev. Lett.  [**83**]{}, 1914 (1999) \[arXiv:hep-ph/9905312\];\
M. Beneke, G. Buchalla, M. Neubert and C. T. Sachrajda, Nucl. Phys.  B [**591**]{}, 313 (2000) \[arXiv:hep-ph/0006124\].

C. W. Bauer, D. Pirjol and I. W. Stewart, Phys. Rev.  D [**65**]{}, 054022 (2002) \[arXiv:hep-ph/0109045\];\
C. W. Bauer, D. Pirjol, I. Z. Rothstein and I. W. Stewart, Phys. Rev.  D [**70**]{}, 054015 (2004) \[arXiv:hep-ph/0401188\].

Y. Y. Keum, T. Kurimoto, H. N. Li, C. D. Lu and A. I. Sanda, Phys. Rev.  D [**69**]{}, 094018 (2004) \[arXiv:hep-ph/0305335\]. C. D. Lu, Phys. Rev.  D [**68**]{}, 097502 (2003) \[arXiv:hep-ph/0307040\];\
C. D. Lu and K. Ukai, Eur. Phys. J.  C [**28**]{}, 305 (2003) \[arXiv:hep-ph/0210206\];\
Y. Li and C. D. Lu, J. Phys. G [**29**]{}, 2115 (2003) \[arXiv:hep-ph/0304288\];\
Y. Li and C. D. Lu, HEP&NP. [**27**]{}, 1062 (2003) \[arXiv:hep-ph/0305278\];\
Y. Li, C. D. Lu and Z. J. Xiao, J. Phys. G [**31**]{}, 273 (2005) \[arXiv:hep-ph/0308243\]. Y. Y. Keum, H. n. Li and A. I. Sanda, Phys. Lett.  B [**504**]{}, 6 (2001) \[arXiv:hep-ph/0004004\];\
Y. Y. Keum, H. N. Li and A. I. Sanda, Phys. Rev.  D [**63**]{}, 054008 (2001) \[arXiv:hep-ph/0004173\]. C. D. Lu, K. Ukai and M. Z. Yang, Phys. Rev.  D [**63**]{}, 074009 (2001) \[arXiv:hep-ph/0004213\];\
C. D. Lu and M. Z. Yang, Eur. Phys. J.  C [**23**]{}, 275 (2002) \[arXiv:hep-ph/0011238\].

A. Ali, G. Kramer, Y. Li, C. D. Lu, Y. L. Shen, W. Wang and Y. M. Wang, arXiv:hep-ph/0703162. G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys.  [**68**]{}, 1125 (1996) \[arXiv:hep-ph/9512380\]. C. H. Chen, Phys. Lett.  B [**560**]{}, 178 (2003) \[arXiv:hep-ph/0301154\]. E. Follana, C. T. H. Davies, G. P. Lepage and J. Shigemitsu \[HPQCD Collaboration\], arXiv:0706.1726 \[hep-lat\]. B. Aubert [*et al.*]{} \[BABAR Collaboration\], Phys. Rev.  D [**73**]{}, 112004 (2006) \[arXiv:hep-ex/0604037\].

K. Abe [*et al.*]{} \[Belle Collaboration\], arXiv:0708.1668 \[hep-ex\]. H. Y. Cheng, C. K. Chua and A. Soni, Phys. Rev.  D [**71**]{}, 014030 (2005) \[arXiv:hep-ph/0409317\]; C.D. Lu and Y.L. Shen, Phys. Rev. D73, 034005 (2006).

H. n. Li, S. Mishima and A. I. Sanda, Phys. Rev.  D [**72**]{}, 114005 (2005) \[arXiv:hep-ph/0508041\]. H. n. Li and S. Mishima, Phys. Rev.  D [**74**]{}, 094020 (2006) \[arXiv:hep-ph/0608277\].

[^1]: e-mail: liying@ytu.edu.cn
