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Disc fragmentation plays an important role in determining the number ofprimordial stars (Pop III stars), their masses, and hence the initial massfunction. In this second paper of a series, we explore the effect of uniformFUV H$_2$-photodissociating and X-ray radiation backgrounds on the formation ofPop~III stars using a grid of high-resolution zoom-in simulations. We findthat, in an X-ray background, protostellar discs have lower surface density andhigher Toomre $Q$ parameter, so they are more stable. For this reason, X-rayirradiated discs undergo fewer fragmentations and typically produce eitherbinary systems or low-multiplicity systems. In contrast, the cases with weak orno X-ray irradiation produce systems with a typical multiplicity of $6 \pm 3$.In addition, the most massive protostar in each system is smaller by roughly afactor of two when the disc is irradiated by X-rays, due to lower accretionrate. With these two effects combined, the initial mass function of fragmentsbecomes more top-heavy in a strong X-ray background and is well described by apower-law with slope $1.53$ and high-mass cutoff of $61$ M$_\odot$. WithoutX-rays, we find a slope $0.49$ and cutoff mass of $229$ M$_\odot$. Finally,protostars migrate outward after their formation due to the accretion ofhigh-angular momentum gas from outside and the migration is more frequent andsignificant in absence of X-ray irradiation.

Population III Star Formation in an X-ray background: II. Protostellar Discs, Multiplicity and Mass Function of the Stars