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The random-field XY model is studied in spatial dimensions d=3 and 4, andin-between, as the limit q --> \infty of the q-state clock models, by the exactrenormalization-group solution of the hierarchical lattice or, equivalently,the Migdal-Kadanoff approximation to the hypercubic lattices. Thelower-critical dimension is determined between 3.81 < d_c <4. When therandom-field is scaled with q, a line segment of zero-temperature criticalityis found in d=3. When the random-field is scaled with q^2, a universal phasediagram is found at intermediate temperatures in d=3.

Lower-Critical Dimension of the Random-Field XY Model and the Zero-Temperature Critical Line