  We address the problem of the lattice Stark shifts in the Sr clock caused by
the multipolar $M1$ and $E2$ atom-field interactions and by the term nonlinear
in lattice intensity and determined by the hyperpolarizability. We have
developed an approach to calculate hyperpolarizabilities for atoms and ions
based on a solution of the inhomogeneous equation which allows to effectively
and accurately carry out complete summations over intermediate states. We
applied our method to the calculation of the hyperpolarizabilities for the
clock states in Sr. We also carried out an accurate calculation of the
multipolar polarizabilities for these states at the magic frequency.
Understanding these Stark shifts in optical lattice clocks is crucial for
further improvement of the clock accuracy.
