Precision continuous-wave laser measurement of the $\text{1}^\text{3}\text{S}_\text{1} \to \text{2}^\text{3}\text{S}_\text{1}$ interval in positronium

We report a 4.9\,ppb measurement of the positronium $\text{1}^\text{3}\text{S}_\text{1} \to \text{2}^\text{3}\text{S}_\text{1}$ interval using continuous-wave two-photon laser spectroscopy. The transition is detected via photoionization by the same excitation laser. The resulting positrons are guided to a microchannel plate detector, surrounded by scintillators to detect the annihilation photons in coincidence, thereby reducing the background. A Monte Carlo lineshape simulation, accounting for effects such as the second-order Doppler shift and the AC Stark shift, is used to extract a transition frequency of $1233607224.1(6.0)\,\text{MHz}$, consistent with the previous 2.6\,ppb determination of this transition and with the most recent QED calculations at order $\mathcal{O}(\alpha^7\ln^2(1/\alpha))$, which predict $1233607222.12(58)\,\text{MHz}$. Combining the two measurements gives $1233607218.1(2.8)\,\text{MHz}$, reducing the tension with QED to about $1.4\,\sigma$. We also present a semi-analytical lineshape model of $\text{1}^\text{3}\text{S}_\text{1} \to \text{2}^\text{3}\text{S}_\text{1}$ of positronium, which shows excellent agreement with detailed simulations and is validated by the experimental data. This expands on previous work with stable atoms by incorporating effects such as limited lifetime of the atoms, photoionization and AC Stark shift. The lineshape modelling is also applicable to other unstable systems, such as muonium. This provides a powerful tool for optimizing the experimental parameters and gaining deeper insights without the need for computationally intensive simulations.