  It has been shown that HD molecules can form efficiently in metal-free gas
collapsing into massive protogalactic halos at high redshift. The resulting
radiative cooling by HD can lower the gas temperature to that of the cosmic
microwave background, T_CMB=2.7(1+z)K, significantly below the temperature of a
few 100 K achievable via H_2-cooling alone, and thus reduce the masses of the
first generation of stars. Here we consider the suppression of HD-cooling by UV
irradiation in the Lyman-Werner (LW) bands. We include photo-dissociation of
both H_2 and HD, and explicitly compute the self-shielding and shielding of
both molecules by neutral hydrogen as well as the shielding of HD by H_2. We
use a simplified dynamical collapse model, and follow the chemical and thermal
evolution of the gas, in the presence of a UV background. We find that a LW
flux of J_crit = 1e-22 erg/cm^2/sr/s/Hz is able to suppress HD cooling and thus
prevent collapsing primordial gas from reaching temperatures below 100 K. The
main reason for the lack of HD cooling for J>J_crit is the partial
photo-dissociation of H_2, which prevents the gas from reaching sufficiently
low temperatures (T<150K) for HD to become the dominant coolant; direct HD
photo-dissociation is unimportant except for a narrow range of fluxes and
column densities. Since the prevention of HD-cooling requires only partial H_2
photo-dissociation, the critical flux J_crit is modest, and is below the UV
background required to reionize the universe at redshift z=10-20. We conclude
that HD-cooling can reduce the masses of typical stars only in rare halos
forming well before the epoch of reionization.
