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Status and Prospects of Discrete Symmetries Tests in Positronium Decays with the J-PET Detector

Positronium is a unique laboratory to study fundamental symmetries in the Standard Model, reflection in space ($\mathcal{P}$), reversal in time ($\mathcal{T}$), charge conjugation ($\mathcal{C}$) and their combinations. The experimental limits on the $\mathcal{C}$, $\mathcal{CP}$ and $\mathcal{CPT}$ symmetries violation in the decays of positronium are still several orders of magnitude higher than the expectations. The newly constructed Jagiellonian Positron Emission Tomograph (J-PET) was optimized for the registration of photons from the electron-positron annihilations. It enables tests of discrete symmetries in decays of positronium atoms via the determination of the expectation values of the discrete-symmetries-odd operators. In this article we present the capabilities of the J-PET detector in improving the current precision of discrete symmetries tests and report on the progress of analysis data from the first data-taking runs.