  We present a systematic study of coupled $S=1/2$ Heisenberg antiferromagnetic
chains in an effective staggered field. We investigate several effects of the
staggered field in the {\em higher} ({\em two or three}) {\em dimensional} spin
system analytically. In particular, in the case where the staggered field and
the inter-chain interaction compete with each other, we predict, using
mean-field theory, a characteristic phase transition. The spin-wave theory
predicts that the behavior of the gaps induced by the staggered field is
different between the competitive case and the non-competitive case. When the
inter-chain interactions are sufficiently weak, we can improve the mean-field
phase diagram by using chain mean-field theory and the analytical results of
field theories. The ordered phase region predicted by the chain mean-field
theory is substantially smaller than that by the mean-field theory.
