Remarks on the penguin decay $B_s \to \phi \phi$ with prospects for FCCee

We underline the theoretical interest of the vector-vector penguin decay $B_s \to \phi \phi$, very clean from the experimental point of view. The CP-violation asymmetry $A_{CP}^{mix}$ comes from the interference of mixing and decay $\lambda_{\phi \phi} = {q \over p} {\bar{A} \over A}$. In the Standard Model (SM) and in the Naive Factorization limit, the CP phase from the mixing ${q \over p}$ exactly cancels the CP phase from the decay ratio ${\bar{A} \over A}$. Therefore, this mode is suitable to look for possible New Physics (NP) because $A_{CP}^{mix}$ would directly indicate the departure from the SM in mixing. We estimate the deviation from this cancellation by analyzing possible small effects in the SM, using in particular the QCD Factorization scheme. We compare the theoretical expectation for $A_{CP}^{mix}$ to the measurement of LHCb, and the implications for NP. We pay also special attention to the transverse amplitude $h = -$, the longitudinal and transverse polarization fractions, and the interesting helicity-dependent observables $\lambda_{\phi \phi}^{h=0}$ and $\lambda_{\phi \phi}^{h=-}$. On the other hand, we make an estimation of the expected sensitivity at the future FCCee experiment for the CP phase and modulus of $\lambda_{\phi \phi}$. We find $\delta(\mid \lambda_{\phi \phi}\mid) = 0.004$ and $\delta(\phi_{\phi \phi}) = 0.009$ rad and, comparing to the LHCb data, we point out the expectations at FCCee in the search of NP.