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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 wellinvestigated in the context of supermassive black hole formation in the earlyuniverse. However, the average intensity in the early universe is severalorders of magnitude lower. For a comprehensive study, we investigate theeffects of LW radiation at $18$ different intensities, ranging from $J_{\rmLW}/J_{21}=0$ (no radiation) to $30$ (H-cooling cloud), on the primordialstar-forming gas cloud obtained from a three-dimensional cosmologicalsimulation. The overall trend with increasing radiation intensity is a gradualincrease in the gas cloud temperature, consistent with previous works. Due tothe HD-cooling, on the other hand, the dependence of gas cloud temperature on$J_{\rm LW}$ deviates from the aforementioned increasing trend for a specificrange 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 firstformation 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-massPop III star with $\leq\!0.8\,M_{\odot}$ (so-called "surviving star") couldform. 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 becomeseffective. This tendency indicates that, whereas the total gas mass in the hosthalo increases with the LW radiation intensity, the total Pop III stellar massdoes not increase similarly.

Low-mass Pop III star formation due to the HD-cooling induced by weak Lyman-Werner radiation