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We underline the theoretical interest of the vector-vector penguin decay $B_s\to \phi \phi$, very clean from the experimental point of view. TheCP-violation asymmetry $A_{CP}^{mix}$ comes from the interference of mixing anddecay $\lambda_{\phi \phi} = {q \over p} {\bar{A} \over A}$. In the StandardModel (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 ${\bar{A} \overA}$. Therefore, this mode is suitable to look for possible New Physics (NP)because $A_{CP}^{mix}$ would directly indicate the departure from the SM inmixing. We estimate the deviation from this cancellation by analyzing possiblesmall effects in the SM, using in particular the QCD Factorization scheme. Wecompare the theoretical expectation for $A_{CP}^{mix}$ to the measurement ofLHCb, and the implications for NP. We pay also special attention to thetransverse amplitude $h = -$, the longitudinal and transverse polarizationfractions, and the interesting helicity-dependent observables $\lambda_{\phi\phi}^{h=0}$ and $\lambda_{\phi \phi}^{h=-}$. On the other hand, we make anestimation of the expected sensitivity at the future FCCee experiment for theCP phase and modulus of $\lambda_{\phi \phi}$. We find $\delta(\mid\lambda_{\phi \phi}\mid) = 0.004$ and $\delta(\phi_{\phi \phi}) = 0.009$ radand, comparing to the LHCb data, we point out the expectations at FCCee in thesearch of NP.

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