Contribution of negative-energy states to multipolar polarizabilities of the Sr optical lattice clock

We address the problem of lattice light shifts in the Sr clock caused by multipolar $M1$ and $E2$ atom-field interactions. We present a simple but accurate formula for the magnetic-dipole polarizability that takes into account the contributions of both the positive- and negative-energy states. We calculate the contribution of negative-energy states to the $M1$ polarizabilities of the clock ${}^1{S}_0$ and ${}^3{P}_0^o$ states at the magic frequency. Taking these contributions into account, we obtain good agreement with the experimental results, explaining the major discrepancy between theory and experiment.