Nonlocal-in-time effective one body Hamiltonian in scalar-tensor gravity at third post-Newtonian order

We complete the nonlocal-in-time effective-one-body (EOB) formalism of the conservative dynamics for massless scalar-tensor (ST) theories at third post-Newtonian (PN) order. The nonlocal-in-time EOB Hamiltonian is obtained by mapping the order-reduced Hamiltonian corresponding to the nonlocal-in-time Lagrangian derived in [Phys. Rev. D 99, 044047 (2019)]. To transcribe the dynamics within the EOB formalism, we use a strategy of order-reduction of nonlocal dynamics to a local ordinary action-angle Hamiltonian. We then map this onto the EOB Hamiltonian to determine the nonlocal-in-time ST corrections to the EOB potentials $(A,B,{Q}_{e})$ at 3PN order.