We present a study of the morphology and intensity of star formation in the host galaxies of eight Palomar-Green quasars using observations with the Hubble Space Telescope. Our observations are motivated by recent evidence for a close relationship between black hole growth and the stellar mass evolution in its host galaxy. We use narrow-band [O II] $lambda$3727, H$beta$, [O III] $lambda$5007 and Pa$alpha$ images, taken with the WFPC2 and NICMOS instruments, to map the morphology of line-emitting regions, and, after extinction corrections, diagnose the excitation mechanism and infer star-formation rates. Significant challenges in this type of work are the separation of the quasar light from the stellar continuum and the quasar-excited gas from the star-forming regions. To this end, we present a novel technique for image decomposition and subtraction of quasar light. Our primary result is the detection of extended line-emitting regions with sizes ranging from 0.5 to 5 kpc and distributed symmetrically around the nucleus, powered primarily by star formation. We determine star-formation rates of order a few tens of M$_odot$/yr. The host galaxies of our target quasars have stellar masses of order $10^{11}$ M$_odot$ and specific star formation rates on a par with those of M82 and luminous infrared galaxies. As such they fall at the upper envelope or just above the star-formation mass sequence in the specific star formation vs stellar mass diagram. We see a clear trend of increasing star formation rate with quasar luminosity, reinforcing the link between the growth of the stellar mass of the host and the black hole mass found by other authors.
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