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We investigate the detection prospects for two-neutrino and neutrinolesssecond 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. Wesimulate the decays in a liquid xenon medium and develop a reconstructionalgorithm which uses the multi-particle coincidence in these decays to separatesignal from background. This is used to compute the expected detectionefficiencies as a function of position resolution and energy threshold forplanned experiments. In addition, we consider an exhaustive list of possiblebackground sources and find that they are either negligible in rate or can begreatly 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 upcomingDark Matter experiments (e.g. XENONnT or LZ), and their major background willbe from gamma rays in the detector construction materials. Second, searches forthe $0\nu\text{EC}\beta^+$ decay mode are likely to be background-free, and newparameter space may be within the reach. To this end we investigate threedifferent scenarios of existing experimental constraints on the effectiveneutrino mass. The necessary 500 kg-year exposure of $^{124}$Xe could beachieved by the baseline design of the DARWIN observatory, or by extracting andusing the $^{124}$Xe from the tailings of the nEXO experiment. We demonstratehow 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.

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