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Theoretical estimate of the half-life for the radioactive $^{134}$Cs and $^{135}$Cs in astrophysical scenarios

We analyze the $^{134}_{55}$Cs$\rightarrow^{134}_{56}$Ba and $^{135}_{55}$Cs$\rightarrow^{135}_{56}$Ba $β^-$ decays, which are crucial production channels for Ba isotopes in Asymptotic Giant Branch (AGB) stars. We reckon, from relativistic quantum mechanis, the effects of multichannel scattering onto weak decays, including nuclear and electronic excited states (ES) populated above $\simeq$ 10 keV, for both parent and daughter nuclei. We find increases in the half-lives for $T>10^8$ K (by more than a factor 3 for $^{134}$Cs) as compared to previous works based on systematics. We also discuss our method in view of these previous calculations. An important impact on half-lives comes from nuclear ES decays, while including electronic temperatures yields further increases of about 20\% at energies 10-30 keV, typical of AGB stars of moderate mass ($M \lesssim 8~M_{\odot}$). Despite properly considering these effects, the new rates remain sensitively lower than the TY values, implying longer half-lives at least above 8-9 keV. Our rate predictions are in substantial accord with recent results based on the shell model, and strongly modify branching ratios along the $s$-process path previously adopted. With our new rate, nucleosynthesis models well account for the isotopic admixtures of Ba in presolar SiC grains and in the Sun.