High-Accuracy Calculation of the Blackbody Radiation Shift in the ${}^{133}\mathrm{Cs}$ Primary Frequency Standard

The blackbody radiation (BBR) shift is an important systematic correction for the atomic frequency standards realizing the SI unit of time. Presently, there is controversy over the value of the BBR shift for the primary ${}^{133}\mathrm{Cs}$ standard. At room temperatures, the values from various groups differ at the $3\times {10}^{-15}$ level, while modern clocks are aiming at ${10}^{-16}$ accuracies. We carry out high-precision relativistic many-body calculations of the BBR shift. For the BBR coefficient $\beta$ at $T=300\mathrm{K}$, we obtain $\beta =-(1.710\pm 0.006)\times {10}^{-14}$, 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 semiempirical calculations. We identify an oversight in those calculations.