Violations of $ CPT$ symmetry and Lorentz invariance are searched for by studying interference effects in $ B^0$ mixing and in $ B^0_s$ mixing. Samples of $ B^0\to J/\psi K^0_{\mathrm{S}}$ and $ B^0_s\to J/\psi K^+ K^$ decays are recorded by the LHCb detector in protonproton collisions at centreofmass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb$^{1}$. No periodic variations of the particleantiparticle mass differences are found, consistent with Lorentz invariance and $ CPT$ symmetry. Results are expressed in terms of the Standard Model Extension parameter $\Delta a_{\mu}$ with precisions of $ \mathcal{O}(10^{15})$ and $ \mathcal{O}(10^{14})$ GeV for the $ B^0$ and $ B^0_s$ systems, respectively. With no assumption on Lorentz (non)invariance, the $ CPT$violating parameter $z$ in the $ B^0_s$ system is measured for the first time and found to be $ \mathcal{R}e(z) = 0.022 \pm 0.033 \pm 0.005$ and $ \mathcal{I}m(z) = 0.004 \pm 0.011\pm 0.002$, where the first uncertainties are statistical and the second systematic.
Values of $\mathcal{R}e (z)$ obtained from fits in bins of sidereal phase for (top) the $ B ^0$ sample and (bottom) the $ B ^0_ s $ sample. The solid line shows the variation of $\mathcal{R}e (z)$ from the $\Delta a_{\mu}$ fits, using the average $ B $ momentum. 
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Periodograms for (top) the $ B ^0$ and (bottom) the $ B ^0_ s $ sample. The insets show a zoom around the solar (red dashed line) and sidereal (blue solid line) frequencies, which have been made by highly oversampling the frequencies in this narrow range. 
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Animated gif made out of all figures. 
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Timedependent functions $h_k(t)$ in Eq. 4. 
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Definition of the coefficients in Eq. 5. The following definitions are used: $\eta^+ \equiv (1+\eta_{l}\eta_{m})/2$, $\eta^ \equiv (1\eta_{l}\eta_{m})/2$, $\eta^{\mathcal{I}m } \equiv i(\eta_{l}\eta_{m})/2$, $\eta^{\mathcal{R}e } \equiv (\eta_{l}+\eta_{m})/2$. Furthermore, $\zeta ^{+} \equiv (\zeta )^{\eta^+ }$, and $\zeta ^{} \equiv (\zeta )^{\eta^ }$, such that $\zeta ^{\pm}=1$ if $\eta^{\pm} =0$ and $\zeta ^{\pm}=\zeta $ otherwise. 
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Systematic uncertainties on $\Delta a_{\mu}$ for $ B ^0$ mixing and on $\Delta a_{\mu}$ and $z$ for $ B ^0_ s $ mixing. Contributions marked with  are negligible. 
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Supplementary material full pdf 
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This ZIP file contains supplemetary material for the publication LHCbPAPER2016005. The files are: supplementary.pdf : An overview of the extra figures *.pdf, *.png, *.eps, *.C : The figures in various formats 
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Created on 15 February 2019.Citation count from INSPIRE on 15 February 2019.