PRCS versus quark number density

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May 13, 2009 - it has N pairs of complex conjugate poles at p2 = −(ai ± ibi)2. On the ..... Zong H S, Chang L, Hou F Y, Sun W M and Liu Y X 2005 Phys. Rev.
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Partial restoration of chiral symmetry versus quark number density at zero temperature

This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2009 J. Phys. G: Nucl. Part. Phys. 36 064073 (http://iopscience.iop.org/0954-3899/36/6/064073) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

JOURNAL OF PHYSICS G: NUCLEAR AND PARTICLE PHYSICS

J. Phys. G: Nucl. Part. Phys. 36 (2009) 064073 (6pp)

doi:10.1088/0954-3899/36/6/064073

Partial restoration of chiral symmetry versus quark number density at zero temperature Huan Chen, Wei Yuan and Yu-Xin Liu Department of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871, People’s Republic of China E-mail: [email protected]

Received 26 December 2008 Published 13 May 2009 Online at stacks.iop.org/JPhysG/36/064073 Abstract We calculate the chiral condensate and quark number density with both assumed analytical forms and numerical solution of the Dyson–Schwinger equation of the quark propagator at finite chemical potential. It shows that once the quark number density becomes nonzero, the chiral condensate below the Fermi surface is remarkably reduced and even has its sign flipped. It provides an explanation for the partial restoration of chiral symmetry in QCD matter.

1. Introduction There have been evidences (see, for example, [1] for a review) showing that chiral symmetry, which is dynamically broken in vacuum, can be partially restored in nuclear matter at low temperature and low density. Based on hadron degrees of freedom and effective field theory (e.g. [1, 2]), partial restoration of chiral symmetry (PRCS) has been extensively studied. The Nambu–Jona-Lasino model, based on quark degrees of freedom, also shows that the chiral condensate decreases with the increase of density (see, for instance, [3] for a review). However, due to the ‘sign problem’, lattice QCD still cannot provide enough information about the QCD at finite chemical potential at present. Fortunately, it has been shown that Dyson– Schwinger equations (DSEs) also provide good tools for studying nonperturbative QCD [4–7]. Within this framework, chiral dynamics at finite chemical potentials has been widely studied [6, 8–11] and the variation of the chiral quark condensate is confirmed. Nevertheless, the details of such behavior have not yet been well discussed either. In this work, we will try to shed light on such an issue in the framework of Dyson–Schwinger equations of QCD. 0954-3899/09/064073+06$30.00 © 2009 IOP Publishing Ltd Printed in the UK

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J. Phys. G: Nucl. Part. Phys. 36 (2009) 064073

H Chen et al

2. Chemical potential and analytic continuation Our starting point is the quark propagator in the Euclidean space. Its general structure at zero chemical potential can be written as S0 (p) = Z(p2 )

−iγ · p + M(p2 ) , p2 + M 2 (p2 )

(1)

which can be determined by solving the truncated DSEs. In this work we concentrate on the case of the chiral limit, i.e. with current quark mass m = 0, in which M(p2 ) = 0 represents the dynamical chiral symmetry breaking (DCSB). It has been shown that, below a critical chemical potential μa , the quark propagator can be given by the analytic continuation of that at μ = 0, i.e. ˜ S(p; μ) = S0 (p),

(2)

where p˜ = ( p, p˜ 4 ) = ( p, p4 + iμ). μa can be given by the real part of the first pole (nearest to the original point) of S0 (p) [10]. However, as the first step to investigate the influence of the chemical potential, we simply assume that equation (2) works even above μa in this section. To quantitatively describe the PRCS in matter, we calculate the quark number density nq (μ), ¯ chiral condensate −qq(μ) and their corresponding distribution functions, defined as  nq (μ) = 2Nc Nf Z2 1 p|; μ) = f1 (| 4π



−∞

¯ −qq(μ) = 2Nc Z4 1 4π



(3)





p|; μ) = f2 (|

d3 p f1 (| p|; μ), (2π )3 dp4 trD (−γ4 S(p; μ)),

(4)

d3 p f2 (| p|; μ), (2π )3

(5)

∞ −∞

dp4 trD S(p; μ),

(6)

where the color number Nc = 3, the flavor number Nf = 2, and Z2 and Z4 are the renormalization constants of the quark wavefunction and quark mass, respectively. Since yet little is known about the analytic structure of the quark propagator on the whole complex plane, we will investigate two types of the parameterized quark propagator, with poles and/or branch cuts. First, we assume a meromorphic form of the quark propagator at μ = 0:   N  1 ri 1 , (7) + SN (p) = 2 iγ · p + ai + ibi iγ · p + ai − ibi i=1 where, without loss of generality, |ai | < |ai+1 |, i = 1, . . . , N − 1. On the complex p2 plane, it has N pairs of complex conjugate poles at p2 = −(ai ± ibi )2 . On the complex p4 plane, the poles locate at    2  1 p  2 + ai2 − bi2 + 4ai2 bi2 − p Zi = ±(xi ± iyi ) = ± √2  2 + ai2 − bi2 

±i 2

 2  p  2 + ai2 − bi2 + 4ai2 bi2 + p  2 + ai2 − bi2 .

(8)

J. Phys. G: Nucl. Part. Phys. 36 (2009) 064073

H Chen et al

Figure 1. Numerical results of f20 (| p|), f1 (| p|; μ > μa ) and f2 (| p |; μ > μa ). Left: with a fitted form in equation (7) and parameter set ‘2CC’. Right: with a solution of the DSE in equation (15) with the model in equations (16)–(20).

One can get f1 (| p|; μ = 0) = 0, and using the residue theorem, one has f20 (| p|) ≡ f2 (| p|; μ = 0) =

N 

ri ς (ai )

i=1

|bi |xi + |ai |yi , xi2 + yi2

(9)

where the sign function: ς (x > 0) = 1 and ς (x < 0) = −1. At μ = 0, due to the shift p4 → p˜ 4 + iμ, the poles move to Zi = ±(xi ± iyi ) − iμ, and one then formally gets f1 (| p|; μ) =

N 

θ (μ − |ai |)θ (pf,i − | p|)ri ,

(10)

i=1

f2 (| p|; μ) ≡ f2 (| p|; μ) − f20 (| p|) =−

N  i=1

θ (μ − |ai |)θ (pf,i − | p|)ri ς (ai )

|bi |xi + |ai |yi , xi2 + yi2

(11)

√ 2 2 2 2 (μ −ai )(μ +bi ) and the step function: θ (x > 0) = 1 and θ (x < 0) = 0. where pf,i = μ As an example, we take the set of parameters ‘2CC’: N = 2, r1 = 0.72, (a1 , b1 ) = (0.351, 0.08) GeV, r2 = 0.28, (a2 , b2 ) = (−0.899, 0.463) GeV, taken from [12] by fitting p|), f1 (| p|; μ) and f2 (| p|; μ) are showed lattice QCD data [13]. The results obtained of f20 (| p|; μ) and f2 (| p|; μ), with opposite in the left panel of figure 1. The nonzero value of f1 (| signs, appears simultaneously below the Fermi surface | p| = pf,1 , where the poles of the quark propagator cross the real p4 axis. In more detail, the fact that the absolute value of p|; μ) below the Fermi surface is larger than that of f20 (| p|) indicates that the chiral f2 (| condensate in the momentum channels, which quarks have occupied, is reduced drastically p| = 0; μ) are illustrated in the left panel of and even has its sign flipped. The results of f1 (| figure 2. The decrease of the chiral condensate with respect to the quark number density is displayed in the right panel of figure 2. Second, we investigate the case in which the quark propagator involves branch cuts. The form at μ = 0 is taken from [12], signed as ‘BCfit1’,  α p2 + m2s C2 2 + 2 , (12) Z(p ) = 1 − 2π p + m2s + 2QCD 3

J. Phys. G: Nucl. Part. Phys. 36 (2009) 064073

H Chen et al

Figure 2. (Left) Numerical results of f1 (| p | = 0; μ). (Right) Decreasing feature of the chiral condensate with respect to the quark number density.

M(p ) = Cd 2

1−γm  α p2 + m2s p2

+

m2s

+

2QCD

wherein the strong running coupling [14] p2 α(0) 4π

2 α(p ) = + 2 2 1 + p QCD β0 p + 2QCD 2



C4 + 2 , p2 + m2s + 2QCD

2QCD 1  2 − p2 − 2QCD ln p2 QCD

(13)

,

(14)

11N −2N

c f , β0 = , γm = 12 with Nc = 3, Nf = 0. The parameters where α(0) = 8.915 Nc 3 33 3 are [12] Cd = 0.086 GeV , C4 = 0.248 GeV5 , C2 = −0.011 GeV2 , ms = 0.5 GeV and QCD = 0.9 GeV. This quark propagator has a branch point at time-like momentum p2 = −m2s and a pair of complex conjugate poles at p2 = −M 2 (p2 ). The numerical results displayed in figure 2 show that the branch cuts do not influence the results qualitatively.

3. Results of solving the DSE at finite chemical potential In this section, we drop the assumption of equation (2) and explicitly solve the DSE for the quark propagator at finite chemical potential:  + iγ4 (p4 + iμ)) S(p; μ)−1 = Z2 (iγ · p  d4 q λa + Z1 g 2 (μ) Dρσ (p − q; μ) γρ S(q; μ) σa (q, p; μ), (15) 4 (2π ) 2 where Dρσ (p −q; μ) is the effective gluon propagator, σa (q, p; μ) is the dressed quark–gluon vertex, and Z1 and Z2 are the renormalization constants. Our ans¨atze for Dρσ (p − q; μ) and

σa (q, p; μ) are   G ((p − q)2 ) (p − q)ρ (p − q)σ λa δ

σ , Z1 g 2 Dρσ (p − q; μ) σa (q, p; μ) = − (16) ρσ (p − q)2 (p − q)2 2 with an extended form of the Ball–Chiu vertex at finite chemical potential [10]:   i ⊥ ⊥ ˜ σ γ · (q˜ + p) ˜ A (q, p; μ) i σ (q, p; μ) = iA (q, p; μ)γσ + iC (q, p; μ)γσ + (q˜ + p) 2   i ˜ C (q, p; μ) + B (q, p; μ) , ˜ σ γ · (q˜ + p) (17) + (q˜ + p) 2 4

J. Phys. G: Nucl. Part. Phys. 36 (2009) 064073

H Chen et al

 1 F ( q 2 , q4 ; μ) + F ( p2 , p4 ; μ) , (18) 2 p2 , p4 ; μ) F ( q 2 , q4 ; μ) − F ( , (19) F (q, p; μ) = q˜ 2 − p˜ 2  γ4 ), γ ⊥ = γ − γ , F = A, B, C, and the model-effective interaction where γ = (0, 4π 2 G (s) 2 = 6 Ds e−s/ω . (20) s ω The parameters are ω = 0.5 GeV and D = 12 GeV2 , and the first pole of the quark propagator locates at p2 = −(a1 ± ib1 )2 = −0.2972 GeV2 . p|), f1 (| p|; μ) and f2 (| p|; μ) are displayed on the right panel Numerical results of f20 (| ¯ 1/3 and nq is illustrated in figure 2. of figure 1 and that of the relation between −(−qq) It is apparent that qualitatively similar results as those given by the assumed analytical quark propagator appear. F (q, p; μ) =

4. Summary and discussion In summary, we have discussed the relation between the PRCS and the quark number density in matter. By analyzing the chiral condensate distribution function and quark number density distribution function, we show evidently that, corresponding to the appearance of nonzero quark number density, the chiral condensate, in the momentum channels which quarks have occupied, is remarkably reduced and even has its sign flipped. Such a ‘negative’ condensate below the Fermi surface balances the original one beyond the Fermi surface and causes the integrated chiral condensate to remarkably decrease. Then the PRCS appears. We thus give a mechanism of the PRCS in quark matter. Finally, it is remarkable that nuclear matter is the one with quarks being confined in hadrons. To provide a reasonable mechanism for the PRCS in nuclear matter at low density, one needs to provide enough evidences and even new ingredients for quark confinement in the framework of Dyson–Schwinger equations of QCD. The related investigation is in progress. Acknowledgments This work was supported by the National Natural Science Foundation of China under contract no. 10425521 and no. 10675007, and the Major State Basic Research Development Program under contract no. G2007CB815000. The authors gratefully acknowledge Dr C D Roberts and Dr L Chang for their helpful discussions. References [1] [2] [3] [4] [5]

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