Measurement of two-neutrino double electron capture half-life of $^{124}$Xe with PandaX-4T

Detailed studies of two-neutrino double electron capture (2$ν$DEC) is a crucial step towards searching for the neutrino-less mode to explore the Majorana nature of neutrinos. We have measured precisely the half-life of the 2$ν$DEC process in $^{124}$Xe, utilizing a total exposure of 1.73 tonne$\cdot$year from the commissioning run and the first science run of the PandaX-4T experiment. A time-dependent background model in the $\mathcal{O}$(10 keV) energy is constructed for the first time in PandaX-4T data. With an unbinned maximum likelihood fit, we determine the half-life of the 2$ν$DEC process to be $(1.03\pm0.15_{\rm stat}\pm0.08_{\rm sys})\times 10^{22}$$\,$yr. Furthermore, we have evaluated the branching ratio for both electrons captured from the $K$ shell ($KK$) to be $(65\pm5)\%$, which aligns with the $^{124}$Xe nuclear model calculations within 1.8$\,$$σ$.