Blackbody radiation shift in the Sr optical atomic clock

We evaluated the static and dynamic polarizabilities of the $5s^2~^1\!S_0$ and $5s5p~^3\!P_0^o$ states of Sr using the high-precision relativistic configuration interaction + all-order method. Our calculation explains the discrepancy between the recent experimental $5s^2~^1\!S_0 - 5s5p~^3\!P_0^o$ dc Stark shift measurement $\Delta \alpha = 247.379(7)$ [Middelmann {\it et. al}, Phys. Rev. Lett. {\bf 109}, 263004 (2012)] and the earlier theoretical result of 261(4)~a.u. [Porsev and Derevianko, Phys. Rev. A {\bf 74}, 020502R (2006)]. Our present value of 247.5~a.u. is in excellent agreement with the experimental result. We also evaluated the dynamic correction to the BBR shift with 1\% uncertainty; -0.1492(16)~Hz. The dynamic correction to the BBR shift is unusually large in the case of Sr (7\%) and it enters significantly into the uncertainty budget of the Sr optical lattice clock. We suggest future experiments that could further reduce the present uncertainties.