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

Detailed studies of two-neutrino double electron capture (2$\nu$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$\nu$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$\nu$DEC process to be $(1.03\pm {0.15}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.08}_{\mathrm{s}\mathrm{y}\mathrm{s}})\times {10}^{22}$$$yr. Furthermore, we have evaluated the branching ratio for both electrons captured from the $K$ shell ($KK$) to be $(65\pm 5)\mathrm{\%}$, which aligns with the ${}^{124}$Xe nuclear model calculations within 1.8$$$\sigma$.