Nonderivative modified gravity: A classification

We find a new class of theories of massive gravity with five propagating degrees of freedom where only rotations are preserved. Our results are based on a nonperturbative and background-independent Hamiltonian analysis. In these theories the weak field approximation is well behaved and the static gravitational potential is typically screened a` la Yukawa at large distances, while at short distances no van Dam-Veltman-Zakharov discontinuity is found and there is no need to rely on nonlinear effects to pass the solar system tests. The effective field theory analysis shows that the ultraviolet cutoff is \uf0mMPl e1=2 \u2019 1="m, the highest possible. Thus, these theories can be studied in the weak-field regime at all the phenomenologically interesting scales and are candidates for a calculable large-distance modified gravity