Detection prospects for the second-order weak decays of $^{124}$Xe in multi-tonne xenon time projection chambers

We investigate the detection prospects for two-neutrino and neutrinoless second order weak decays of $^{124}$Xe -- double electron capture ($0/2\nu\text{ECEC}$), electron capture with positron emission ($0/2\nu\text{EC}\beta^+$) and double-positron emission ($0/2\nu\beta^+\beta^+$) -- in multi-tonne xenon time projection chambers. We simulate the decays in a liquid xenon medium and develop a reconstruction algorithm which uses the multi-particle coincidence in these decays to separate signal from background. This is used to compute the expected detection efficiencies as a function of position resolution and energy threshold for planned experiments. In addition, we consider an exhaustive list of possible background sources and find that they are either negligible in rate or can be greatly reduced using our topological reconstruction criteria. In particular, we draw two conclusions: First, with a half-life of $T_{1/2}^{2\nu\text{EC}\beta^+} = (1.7 \pm 0.6)\cdot 10^{23}\,\text{yr}$, the $2\nu\text{EC}\beta^+$ decay of $^{124}$Xe will likely be detected in upcoming Dark Matter experiments (e.g. XENONnT or LZ), and their major background will be from gamma rays in the detector construction materials. Second, searches for the $0\nu\text{EC}\beta^+$ decay mode are likely to be background-free, and new parameter space may be within the reach. To this end we investigate three different scenarios of existing experimental constraints on the effective neutrino mass. The necessary 500 kg-year exposure of $^{124}$Xe could be achieved by the baseline design of the DARWIN observatory, or by extracting and using the $^{124}$Xe from the tailings of the nEXO experiment. We demonstrate how a combination of $^{124}$Xe results with those from $0\nu\beta^-\beta^-$ searches in $^{136}$Xe could help to identify the neutrinoless decay mechanism.