How to separate different CP eigenstates?

Question

Someone can tell me how to separate different CP eigenstates(even and odd) using angular distribution of B meson decay?

Answer 1

I realise I'm responding to an old question, but perhaps you're still in need of an answer.

I'm going to use $B^0_s \to \phi \phi$ as an example. It is a transition of a pseudoscalar meson to two vector mesons. There are 3 possible configurations for angular momentum, $\ell$: aligned, anti-aligned or perpendicular. In the LHCb angular analysis of this decay, polarisation amplitudes are assigned to each: $A_0$ for $\ell=0$, $A_\perp$ for $\ell=1$ and $A_\parallel$ for $\ell=2$. There are also contributions from decays of scalar mesons to the final state, which are assigned amplitudes $A_S$ and $A_{SS}$ for single and double $S$-wave, respectively. Parity is $(-1)^\ell$ so you can see that it depends on whether $\ell$ is odd or even. For this decay, two of the amplitudes, $A_0$ and $A_\parallel$ are CP-even, and $A_\perp$ is CP-odd, hence the final state is a superposition of CP eigenstates.

Each of the five polarisation amplitudes has some angular distribution and time-dependence associated with it. The total amplitude is the sum of these: \begin{equation} A(t,\theta_1,\theta_2,\Phi) = A_0(t)\cos\theta_1\cos\theta_2 + \frac{A_\parallel(t)}{\sqrt{2}}\sin\theta_1\sin\theta_2\cos\Phi + \,i\frac{A_\perp(t)}{\sqrt{2}}\sin\theta_1\sin\theta_2\sin\Phi + \frac{A_S(t)}{\sqrt{2}}(\cos\theta_1+\cos\theta_2)+\frac{A_{SS}(t)}{3} \end{equation}

One can write down a 4D PDF containing 15 terms to fit to the full time-dependent differential decay rate, which takes the following form:

\begin{equation} \frac{\text{d}^4\Gamma}{\text{d}t\;\text{d}\cos\theta_1\;\text{d}\cos\theta_2\;\text{d}\Phi}\propto\sum^{15}_{i=1} K_i(t)f_i(\theta_1,\theta_2,\Phi) \end{equation}

As a side note, the CP-violating phase $\phi^{s\bar{s}s}_s$ (acquired in the interference between mixing and decay, since the final state is accessible from the $B^0_s$ and the $\bar{B^0_s}$) and the "direct"-CPV parameter $|\lambda| = \left|(q/p)(\bar{A}_f/A_f)\right|$ are contained within the $K_i$ terms, which can be decomposed as: \begin{equation} K_i(t) = N_i e^{-\Gamma_s t} \Big[ a_i\cosh\left(\Delta\Gamma_s t/2\right) + b_i\sinh\left(\Delta\Gamma_s t/2\right) + c_i\cos(\Delta m_s t) + d_i\sin(\Delta m_s t) \Big] \end{equation} Where $N_i$ are products of the absolute amplitudes $|A|$ at $t$=0, the coefficients $a_i$, $b_i$, $c_i$ and $d_i$ are functions of $|\lambda|$, $\phi_s$ and the CP-conserving strong phases $\delta$ associated with each amplitude.

Source: Phys. Rev. D 90, 052011 (2014)

http://arxiv.org/abs/1407.2222