6 P = n 1+2 /d 2 m ( d/ 2 + 1) C 2 /d d . (31) Acknowledgments V. CONCLUSIONS AND DISCUSSIONS We notice that Eqs. (30) and (31) are identical to the Fermi energy and the equation of state at T = 0 for a free Fermi gas with mass m ∗ = ( d C $_{d}$/K$_{d}$ ) 2 $^{/d}$m . That is, within the ϵ expansion, we have shown that the thermo- dynamics at T = 0 for a unitary dilute Bose gas is identi- cal to the one for a free Fermi gas. This correspondence between the thermodynamics at T = 0 for the dilute FR Bose gas and the free Fermi gas was also noticed in Refs. 8 and 9 from an entirely different approach. Therefore, such a “fermionization” phenomenon should not be an artifact of either approach, and deserves further investi- gations. We study the ground-state properties of a dilute FR Bose gas from the point of view that this system can be described by a nontrivial zero-density fixed point. From a RG analysis, we first show that the FR Bose gas will always collapse in the dilute limit. Following from this consequence, the MSF state cannot exist in the dilute limit while the ASF state can survive with a finite life- time if we prepare this system from the side with a pos- itive scattering length. Hence, the low-energy properties of a dilute FR Base gas in the unitary regime are sup- posed to be described by the ASF state. Based on this observation, we show that the equilibrium properties of a dilute FR Bose gas at T = 0 exhibit universal scaling forms near the Feshbach resonance. Most of previous works were concentrated on the mean-field phase diagram. In the present paper, we focus on the possible universal properties of this system near the unitary limit. This point was less studied before, and our RG analysis based on the ϵ expansion may shed light on this aspect. In particular, unlike some of previous works $^{9,16}$, the present approach does not rely on the scal- ing hypothesis. In stead, the universality in the unitary regime naturally arises from the presence of a nontrivial zero-density fixed point. Moreover, the roles of danger- ously irrelevant operators, like the u$_{α}$ terms in the action, are more easily captured within the RG framework. The work of Y.-W. Lee is supported by the National Science Council of Taiwan under grant NSC 96-2112-M- 029-006-MY3. The work of Y.L. Lee is supported by the National Science Council of Taiwan under grant NSC 98- 2112-M-018-003-MY3. Our results are obtained by neglecting the higher-order correlations such as the three-body effects $^{17}$. Therefore, they are applicable before the Efimov physics fully sets in $^{18,19}$. It is interesting to see how the three-body effects affect the present predictions. The quantum phase transition between the ASF and MSF phases predicted by the mean-field theory 4 can be observed only when these phases are thermodynamically stable. According to the present analysis (Fig. 1), the ASF and MSF states can be stabilized either by strong dimer-dimer repulsions or by increasing the density of atoms to a moderate value. Therefore, to observe such a quantum phase transition, we must go beyond the dilute limit. A striking result following from our RG analysis is that the T = 0 thermodynamics of this system in the unitary limit is identical to that for a free Fermi gas (with a dif- ferent mass). This point was also pointed out in Refs. 8 and 9 from the variational wavefunction approach. Es- pecially, the chemical potential at zero temperature is of the form µ = αn 2 / $^{3}$/m in the unitary limit, where α takes a universal value. The values of α given by Refs. 8 and 9 are 22 . 22 and 6 . 077, respectively. Since the latter is much smaller than the former, the authors of Ref. 9 claimed that the trial wavefunction adopted by them should be an energetically better candidate for the ground state. By setting d = 3 in Eq. (30), we obtain α = 4 . 996. The fact that the obtained numerical value is close to the one in Ref. 9 provides an evidence which further supports our approximate solutions based on the ϵ ex- pansion. However, to determine the exact value of α , a more comprehensive numerical calculation is warranted. Moreover, it deserves further study whether or not such a correspondence between the dilute FR Bose gas in the unitary limit and the free Fermi gas can be extended to finite temperature and non-equilibrium processes. ∗ Electronic address: yllee@cc.ncue.edu.tw † Electronic address: ywlee@thu.edu.tw 1 C.C. Bradley, C.A. Sackett, J.J. Tollett, and R.G. Hulet, Phys. Rev. Lett. 75 , 1687 (1995); C.A. Sackett, J.M. Ger- ton, M. Welling, and R.G. Hulet, ibid. , 82 , 876 (1999); E.A. Donley, N.R. Claussen, S.L. Cornish, J.L. Roberts, E.A. Cornell, and C.E. Wieman, Nature (London) 412 , 295 (2001); J.L. Roberts, N.R. Claussen, S.L. Cornish, E.A. Donley, E.A. Cornell, and C.E. Wieman, Phys. Rev. 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