We report the results of a comprehensive study of the relationship between galaxy size, stellar mass and specific star-formation rate (sSFR) at redshifts 1.3<z<1.5. Based on a mass complete (M_star >= 6x10^10 Msun), spectroscopic sample from the UKIDSS Ultra-deep Survey (UDS), with accurate stellar-mass measurements derived from spectro photometric fitting, we find that at z~1.4 the location of massive galaxies on the size-mass plane is determined primarily by their sSFR. At this epoch we find that massive galaxies which are passive (sSFR <= 0.1 Gyr^-1) follow a tight size-mass relation, with half-light radii a factor f=2.4+/-0.2 smaller than their local counterparts. Moreover, amongst the passive sub-sample we find no evidence that the off-set from the local size-mass relation is a function of stellar population age. Based on a sub-sample with dynamical mass estimates we also derive an independent estimate of f=2.3+/-0.3 for the typical growth in half-light radius between z~1.4 and the present day. Focusing on the passive sub-sample, we conclude that to produce the necessary evolution predominantly via major mergers would require an unfeasible number of merger events and over populate the high-mass end of the local stellar mass function. In contrast, we find that a scenario in which mass accretion is dominated by minor mergers can produce the necessary evolution, whereby an increase in stellar mass by a factor of ~2, accompanied by an increase in size by a factor of ~3.5, is sufficient to reconcile the size-mass relation at z~1.4 with that observed locally. Finally, we note that a significant fraction (44+/-12%) of the passive galaxies in our sample have a disk-like morphology, providing additional evidence that separate physical processes are responsible for the quenching of star-formation and the morphological transformation of massive galaxies (abridged).
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