Paring density waves as the origin of a ring-like RIXS profile in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

The coexistence of a homogeneous d-wave gap and short-ranged pairing density waves (PDW) accounts for the apparent “ring” charge order in all directions of the copper oxide plane, observed by recent RIXS measurements in Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$