  We carry out an analytical study of quantum spin ice, a U$(1)$ quantum spin
liquid close to the classical spin ice solution for an effective spin $1/2$
model with anisotropic exchange couplings $J_{zz}$, $J_{\pm}$ and $J_{z\pm}$ on
the pyrochlore lattice. Starting from the quantum rotor model introduced by
Savary and Balents in Phys. Rev. Lett. 108, 037202 (2012), we retain the
dynamics of both the spinons and gauge field sectors in our treatment. The
spinons are described by a bosonic representation of quantum XY rotors while
the dynamics of the gauge field is captured by a phenomenological Hamiltonian.
By calculating the one-loop spinon self-energy, which is proportional to
$J_{z\pm}^2$, we determine the stability region of the U$(1)$ quantum spin
liquid phase in the $J_{\pm}/J_{zz}$ vs $J_{z\pm}/J_{zz}$ zero temperature
phase diagram. From these results, we estimate the location of the boundaries
between the spin liquid phase and classical long-range ordered phases.
