A measurement of the timedependent CPviolating asymmetry in $B_s^0\rightarrow\phi\phi$ decays is presented. Using a sample of protonproton collision data corresponding to an integrated luminosity of $5.0$fb$^{1}$ collected by the $ LHCb$ experiment at centreofmass energies $\sqrt{s} = 7$ TeV in 2011, 8 TeV in 2012 and 13 TeV in 2015 and 2016, a signal yield of around 9000 $B_s^0\rightarrow\phi\phi$ decays is obtained. The CPviolating phase $\phi_s^{s\bar{s}s}$ is measured to be $0.073 \pm 0.115$(stat)$\pm 0.027$(syst) rad, under the assumption it is independent on the helicity of the $\phi\phi$ decay. In addition, the CPviolating phases of the transverse polarisations under the assumption of CP conservation of the longitudinal phase are measured. The helicityindependent direct CPviolation parameter is also measured, and is found to be $\lambda=0.99 \pm 0.05 $(stat)$ \pm 0.01 $(syst). In addition, $T$odd tripleproduct asymmetries are measured. The results obtained are consistent with the hypothesis of CP conservation in $b\rightarrow\bar{s}s\bar{s}$ transitions. Finally, a limit on the branching fraction of the $B^0\rightarrow \phi\phi$ decay is determined to be $\mathcal{B}(B^0\rightarrow \phi\phi)<2.7\times 10^{8}$ at 90 confidence level.
A fit to the fourkaon mass for the (top left) 2011, (top right) 2012, (bottom left) 2015 and (bottom right) 2016 data sets, which are represented by the black points. Also shown are the results of the total fit (blue solid line), with the $ B ^0_ s \rightarrow \phi\phi$ (red dashed), the $\Lambda ^0_ b \rightarrow \phi p K ^ $ (magenta long dashed), and the combinatorial (blue short dashed) fit components. 
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Decay angles for the $ B ^0_ s \rightarrow \phi \phi$ decay, where $\theta_{1,2}$ is the angle between the $K^+$ momentum in the $\phi_{1,2}$ meson rest frame and the $\phi_{1,2}$ momentum in the $ B ^0_ s $ rest frame, $\Phi$ is the angle between the two $\phi$ meson decay planes and $\hat{n}_{V_{1,2}}$ is the unit vector normal to the decay plane of the $\phi_{1,2}$ meson. 
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Angular acceptance normalised to the average obtained using simulated $ B ^0_ s \rightarrow \phi\phi$ decays (topleft) integrated over $\cos\theta_2$ and $\Phi$ as a function of $\cos\theta_1$, (topright) integrated over $\cos\theta_1$ and $\Phi$ as a function of $\cos\theta_2$, and (bottom) integrated over $\cos\theta_1$ and $\cos\theta_2$ as a function of $\Phi$. Each figure includes the resulting fit curve. 
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Decaytime acceptances calculated from (left) $ B ^0_ s \rightarrow D ^_ s \pi ^+ $ decays to match Run 1 data and (right) $ B ^0 \rightarrow { J \mskip 3mu/\mskip 2mu\psi \mskip 2mu} K ^{*0} $ decays to match Run 2 data. Superimposed is a parameterisation using cubic splines.. 
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Onedimensional projections of the $ B ^0_ s \rightarrow \phi\phi$ fit for (topleft) decay time with binned acceptance, (topright) helicity angle $\Phi$ and (bottomleft and bottomright) cosine of the helicity angles $\theta_1$ and $\theta_2$. The backgroundsubtracted data are marked as black points, while the blue solid lines represent the projections of the fit. The $ C P$ even $P$wave, the $ C P$ odd $P$wave and the combined $S$wave and double $S$wave components are shown by the red long dashed, green short dashed and purple dotdashed lines, respectively. Fitted components are plotted taking into account efficiencies in the time and angular observables. 
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Fit to the fourkaon invariant mass. The total PDF as described in the text is shown as a blue solid line, $ B ^0_ s \rightarrow \phi\phi$ as a red dashed line, $ B ^0 \rightarrow \phi \phi$ as a green dotted line, the $\Lambda ^0_ b \rightarrow \phi p K$ contribution as a magenta longdashed line and the combinatorial background as a blue shortdashed line. 
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Results of the ${\rm CL}_s$ scan as a function of the $ B ^0 \rightarrow\phi\phi$ yield. The solid black line shows the observed ${\rm CL}_s$ distribution, while the dotted black line indicates the expected distribution. The green (yellow) band marks the $1\sigma$ ($2\sigma$) confidence region on the expected ${\rm CL}_s$. The $90 \%$ CL limit is shown as a red line. 
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Animated gif made out of all figures. 
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Angular coefficients, written in the same basis as the efficiency parameterisation. 
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Tagging performance of the oppositeside (OS) and sameside kaon (SSK) flavour taggers for the $ B ^0_ s \rightarrow \phi\phi$ decay. 
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Results of the decaytimedependent, polarisationindependent fit for the $ C P$ violation fit. Uncertainties shown do not include systematic contributions. 
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Results of the polarisationdependent fit for the $ C P$ violation fit. Uncertainties shown do not include systematic contributions. 
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Summary of systematic uncertainties (in units of $10^{2}$) for parameters of interest in the decaytimedependent measurement. 
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Summary of systematic uncertainties on $A_U$ and $A_V$. 
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Coefficients of the timedependent terms and angular functions used in Eq. ???. Amplitudes are defined at $t=0$. 
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Statistical correlation matrix of the timedependent fit. 
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Statistical correlation matrix of the timedependent fit in which $ C P$ violation is polarisation dependent. 
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Created on 20 February 2021.