  We investigate the Sp(N) mean-field theory for frustrated quantum magnets.
First, we establish some general properties of its solutions; in particular,
for small spin we propose simple rules for determining the saddle points of
optimal energy. We then apply these insights to the pyrochlore lattice. For
spins on a single tetrahedron, we demonstrate a continuous ground state
degeneracy for any value of the spin length. For the full pyrochlore lattice,
this degeneracy translates to a large number of near-degenerate potential
saddle points. Remarkably, it is impossible to construct a saddle point with
the full symmetry of the Hamiltonian--at large N, the pyrochlore magnet CANNOT
be a spin liquid. Nonetheless, for realistic finite values of N, tunnelling
between the nearly degenerate saddle points could restore the full symmetry of
the Hamiltonian.
