Quark-Number Susceptibility at Finite Chemical Potential and Zero Temperature

where Z (µ, T ) is the partition function of QCD at finite temperature and chemical potential. Just as
∗ Supported in part by the National Natural Science Foundation of China under Grant No 10575050, and the Research Fund for the Doctoral Programme of Higher Education in China under Grant No 20060284020. ∗∗ Email: zonghs@ c 2008 Chinese Physical Society and IOP Publishing Ltd
+∞ n=−∞
d3 p (2π )3 (3)
· tr[G[µ, T ](p, ωn )γ4 ],
(2)
where ωn = (2n + 1)πT are the fermion Matsubara frequencies, Nc and Nf denote the number of colours and flavors, respectively, and the trace operation is over Dirac indice. From Eq. (3) it can be seen that the quark-number density ρ(µ, T ) is totally determined by the dressed quark propagator G[µ, T ](p, ωn ) at finite chemical potential and temperature. If one substitutes the free quark propagator at finite T and µ into Eq. (3), it will yield exactly the Fermi statistics result for the quark-number density of a free quark gas, as was described in Ref. [17]. Substituting Eq. (3) into Eq. (2), one can obtain a general formula for the quark-number susceptibility at finite T and µ. As far as we know, the previous works on the quark-number susceptibility mainly treat the case of finite temperature and zero chemical potential, while the treatment of the case of zero temperature and finite chemical potential seems to be scant. So in this paper we concentrate on this case. Setting T = 0 in Eqs. (2) and (3), one obtains ρ(µ) = − Nc Nf d4 p tr[G[µ](p)γ4 ], (2π )4 (4)
PACS: 12. 38. Aw, 12. 38. Lg, 12. 39. −x, 24. 85. +p Analysis of fluctuations is a powerful method for characterizing the thermodynamical properties of a system. It is believed that enhanced fluctuations are an essential characteristic of phase transitions.[1−15] A measure of the intrinsic statistical fluctuations in a system close to thermal equilibrium is provided by the corresponding susceptibilities. In particular, it was recently argued that the quark-number susceptibility χ may be used to identify the chiral critical point in the QCD phase diagram.[7−9] Therefore, it is quite important to achieve a complete understanding of the quark-number susceptibility at finite temperature and/or quark chemical potential (for an isospin symmetric and electrically neutral system the thermodynamical ensemble depends only on the temperature T and the quark chemical potential µ). In this Letter, we try to give a direct method for calculating the quark-number susceptibility at finite chemical potential and zero temperature. The quark-number susceptibility is analogous to the familiar magnetic susceptibility. If we confine ourselves to a two-flavor world with exact isospin symmetry and set µu = µd = µ (µu and µd are the chemical potential of the up and down quark, respectively), the quark-number density and the corresponding susceptibility are given by ρ(µ, T ) = and χ(µ, T ) = T ∂ ln Z (µ, T ) , V ∂µ ∂ρ(µ, T ) , ∂µ (1) expressed by Eq. (2), the quark-number susceptibility measures the response of the number density to an infinitesimal change of the quark chemical potential. From Eq. (1) and using functional integral techniques, one can derive a well-known result (see, for example, Refs. [16,17]) ρ(µ, T ) = − Nc Nf 1 β
NHale Waihona Puke . 2HE Deng-Ke et al.
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χ(µ) =
∂ρ(µ) ∂µ ∂ ∂µ d4 p tr[G[µ](p)γ4 ] , (2π )4 (5)
= − Nc Nf
For simplicity, we choose the model dressed quark propagator proposed in Ref. [27] as our G(p), which reads 4m3 A(p2 ) = 1, B (p2 ) = 2D (9) p with mD = 0.244 GeV. This model propagator has the merits of giving analytic results for the quark-number density and quark-number susceptibility, as will be shown in the following. Here it should be noted that the above dressed quark propagator differs from the most recent result for the quark propagator in lattice calculations.[28] However, in the calculation of some properties of QCD vacuum (such as the QCD vacuum susceptibility) this propagator gives the results consistent with those obtained from more sophisticated quark propagators (see, e.g., Ref. [29]). The qualitative features of the propagator (9) is plain (more detail can be found in Ref. [27]). Here we note that this propagator has an infrared singularity and this infrared enhancement is a signal for dynamical chiral symmetry breaking. Substituting A(p2 ) and B (p2 ) in Eq. (9) into Eq. (8), one has F (p4 ) =