Mixing CP asymmetry calculation

[Valeriia Lukashenko]

Homework Statement

Consider decay $\bar{B^0_s}\rightarrow D^+_s \pi^-$ and calculate its mixing-induces CP asymmetry.

Homework Equations

$$\xi_f^{s}=e^{i\theta_{M_{12}}} \frac{A(\bar{B^0_q}\rightarrow f)}{A(B^0_q \rightarrow f)}=\pm e^{-i\phi_s}\frac{ e^{i\phi_1}|A_1|e^{i\delta_1}+e^{i\phi_2}|A_2|e^{i\delta_2}}{e^{-i\phi_1}|A_1|e^{i\delta_1}+e^{-i\phi_2}|A_2|e^{i\delta_2}}$$

$$\phi_s=2\arg(V^*_{ts}V_{tb})$$

$\phi_1, \phi_2$ are CKM phases.

The Attempt at a Solution

[/B]
I could easily draw diagram for $\bar{B^0_s}\rightarrow D^+_s \pi^-$, but I got stuck on drawing $B^0_s \rightarrow D^+_s \pi^-$. I couldn't find this decay in pdg. But I need it to evaluate $\xi_f^{s}$. How should I evaluate $B^0_s\rightarrow D^+_s \pi^-$ amplitude then?

 

[mfb]

There is a diagram with a second virtual W and a virtual quark, for example.

 

[Valeriia Lukashenko]

I guess you were referring to penguin topologies, but I don't get how to draw properly, so I did this diagram, which has virtual quark for sure (I don't know how to understand if W are virtual). Is it okay? I haven't seen such diagrams so far.

then my both decays have different topologies. Do I still have the right to parametrize my amplitudes in form $A = e^{i \phi_{CKM}} |A_f| e^{i\delta_f}$?

 

[mfb]

There is not enough mass to make a real W, and no W is an external line - they are all virtual.

There is at least one more diagram of this type. It is an odd decay to look at. With a kaon instead of a pion it is studied.

The diagrams don’t have loops - no penguin diagram.