High-accuracy calculation of black-body radiation shift in $^{133}$Cs primary frequency standard*

Black-body radiation (BBR) shift is an important systematic correction for the atomic frequency standards realizing the SI unit of time. In recent years there has been a controversy over the value of the BBR shift for the primary $^{133}$Cs standard. At room temperatures, reported values from various groups have differed at the $3 \times 10^{-15}$ level, while modern clocks are aiming at $10^{-16}$ accuracies. We have carried out high-precision relativistic many-body calculations of the BBR shift. For the BBR coefficient $\beta$ at $T=300K$ we have obtained $\beta=-(1.708\pm0.006) \times 10^{-14}$, implying $6 \times 10^{-17}$ fractional uncertainty. While in accord with the most accurate measurement, our 0.35\%-accurate value is in a substantial, 10\%, disagreement with recent semi-empirical calculations. We have identified an oversight in those calculations, largely resolving the controversy. These results were presented in PRL 97, 040801 (2006).