  Lyman-Werner (LW) radiation photodissociating molecular hydrogen (H$_2$)
influences the thermal and dynamical evolution of the Population III (Pop III)
star-forming gas cloud. The effect of powerful LW radiation has been well
investigated in the context of supermassive black hole formation in the early
universe. However, the average intensity in the early universe is several
orders of magnitude lower. For a comprehensive study, we investigate the
effects of LW radiation at $18$ different intensities, ranging from $J_{\rm
LW}/J_{21}=0$ (no radiation) to $30$ (H-cooling cloud), on the primordial
star-forming gas cloud obtained from a three-dimensional cosmological
simulation. The overall trend with increasing radiation intensity is a gradual
increase in the gas cloud temperature, consistent with previous works. Due to
the HD-cooling, on the other hand, the dependence of gas cloud temperature on
$J_{\rm LW}$ deviates from the aforementioned increasing trend for a specific
range of intensities ($J_{\rm LW}/J_{21}=0.025-0.09$). In HD-cooling clouds,
the temperature remained below $200$ K during $10^5$ yr after the first
formation of the high-density region, maintaining a low accretion rate.
Finally, the HD-cooling clouds have only a low-mass dense core (above
$10^8\,{\rm cm^{-3}}$) with about $1-16\, M_{\odot}$, inside which a low-mass
Pop III star with $\leq\!0.8\,M_{\odot}$ (so-called "surviving star") could
form. The upper limit of star formation efficiency $M_{\rm core}/M_{\rm vir,
gas}$ significantly decreases from $10^{-3}$ to $10^{-5}$ as HD-cooling becomes
effective. This tendency indicates that, whereas the total gas mass in the host
halo increases with the LW radiation intensity, the total Pop III stellar mass
does not increase similarly.
