Can <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si40.svg" display="inline" id="d1e156"><mml:mrow><mml:mi>d</mml:mi><mml:mi>d</mml:mi></mml:mrow></mml:math> excitations mediate pairing?

The Cu-$3d$ states in the high-$T_c$ cuprates are often described as a single band of $3d_{x^2-y^2}$ states, with the other four $3d$ states having about 2 to 3 eV higher energy due to the lower-than-octahedral crystal field at the copper sites. However, excitations to these higher energy states observed with RIXS show indications of strong coupling to doped holes in the $3d_{x^2-y^2}$ band. This relaunches a decades-old question of the possible role of the orbital degrees of freedom that once motivated Bednorz and M\"uller to search for superconductivity in these systems. Here we explore a direction different from the Jahn-Teller electron-phonon coupling considered by Bednorz and M\"uller, namely the interaction between holes mediated by $dd$ excitations.