  We consider possible exotic ground states of quantum spin ice as realized in
rare earth pyrochlores. Prior work in Phys. Rev. Lett. 108, 037202 introduced a
gauge mean field theory (gMFT) to treat spin or pseudospin Hamiltonians for
such systems, reformulated as a problem of bosonic spinons coupled to a U(1)
gauge field. We extend gMFT to treat the most general, nearest neighbor
exchange Hamiltonian, which contains a further exchange interaction, not
considered previously. This term leads to interactions between spinons, and
requires a significant extension of gMFT, which we provide. As an application,
we focus especially on the non-Kramers materials Pr2TM2O7 (TM=Sn, Zr, Hf, and
Ir), for which the additional term is especially important, but for which an
Ising-planar exchange coupling discussed previously is forbidden by
time-reversal symmetry. In this case, when the planar XY exchange is
unfrustrated, we perform a full analysis and find three quantum ground states:
a U(1) quantum spin liquid (QSL), an antiferro-quadrupolar ordered state and a
non-coplanar ferro-quadrupolar ordered one. We also consider the case of
frustrated XY exchange, and find that it favors a pi-flux QSL, with an emergent
line degeneracy of low energy spinon excitations. This feature greatly enhances
the stability of the QSL with respect to classical ordering.
