  Ground state energies for plaquette and dimer order in the $J_1-J_2$
square-lattice spin-half Heisenberg model are compared using series expansion
methods. We find that these energies are remarkably close to each other at
intermediate values of $J_2/J_1$, where the model is believed to have a quantum
disordered ground state. They join smoothly with those obtained from the Ising
expansions for the 2-sublattice N\'eel-state at $J_2/J_1 \approx 0.4$,
suggesting a second order transition from a N\'eel state to a quantum
disordered state, whereas they cross the energy for the 4-sublattice ordered
state at $J_2/J_1 \approx 0.6$ at a large angle, implying a first order
transition to the 4-sublattice magnetic state. The strongest evidence that the
plaquette phase is not realized in this model comes from the analysis of the
series for the singlet and triplet excitation spectra, which suggest an
instability in the plaquette phase. Thus, our study supports the recent work of
Kotov et al, which presents a strong picture for columnar dimer order in this
model. We also discuss the striped nature of spin correlations in this phase,
with substantial resonance all along columns of dimers.
