Strong Coupling Limit/Region of Lattice QCD

1 Strong coupling limit/region of lattice QCD arXiv:0704.2823v1 [nucl-th] 21 Apr 2007 A. Ohnishi, N. Kawamoto, K. Miura, K. Tsubakihara, H. Maekawa Department of Physics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan We study the phase diagram of quark matter and nuclear properties based on the strong coupling expansion of lattice QCD. Both of baryon and finite coupling correction are found to have effects to extend the hadron phase to a larger µ direction relative to Tc . In a chiral RMF model with logarithmic sigma potential derived in the strong coupling limit of lattice QCD, we can avoid the chiral collapse and normal and hypernuclei properties are well described. §1. Introduction Understanding the properties of nuclei and nuclear matter from QCD is one of the ultimate goals in nuclear physics. In a standard roadmap, it is necessary to describe hadrons in QCD, to derive the bear nucleon-nucleon interaction, to obtain the effective nuclear force, and to solve nuclear many-body problems. An alternative way would be to obtain the effective potential (free energy density) in QCD, to represent this effective potential as the density functional in hadronic degrees of freedom, and to apply this density functional to nuclear many-body problems. For this purpose, the most instructive approach may be to combine the strong coupling limit (SCL) of lattice QCD1)–4) and the relativistic mean field (RMF) models, since Monte-Carlo simulations of lattice QCD for dense matter5) are not yet easy at present. In this proceedings, we discuss the baryon and finite coupling effects on the phase diagram in the strong coupling region of lattice QCD,4), 6) and nuclear properties in a chiral RMF model based on the SCL effective potential.7), 8) §2. Effective potential at strong coupling of lattice QCD The strong coupling limit of lattice QCD (SCL-LQCD) predicts the high T second order chiral phase transition at µ = 0 and the high density first order transition at T = 0, and it well explains hadron masses. While these predictions explain the real world qualitatively, quantitative understanding of the phase diagram is not achieved yet. For example, the ratio RµT = µc (T = 0)/Tc (µ = 0) should be larger than two, but it is less than 1/3 in SCL-LQCD without baryon effects. In this work, we discuss the effects of baryons4) and finite coupling on the shape of the phase boundary. At strong coupling (g ≫ 1), the plaquett contribution (∝ 1/g2 ) is perturbative and the effective action is obtained by integrating spatial links as,2) √ S = SSCL + ∆Sg + O(1/d, 1/g 2 d, 1/g4 ) ,  1 X 1 X µ e Vx − e−µ Vx† − (M, VM M ) − (B̄, VB B) + m0 Mx , SSCL = 2 x 2 x (2.1) (2.2) typeset using P T P TEX.cls hVer.0.9i 2 A. Ohnishi, N. Kawamoto, K. Miura, K. Tsubakihara, H. Maekawa ∆Sg = βt X † βs (Vx Vx+ĵ + Vx† Vx−ĵ ) − 2d d−1 x,j>0 X Mx Mx+ĵ Mx+k̂ Mx+k̂+ĵ , (2.3) x,k>j>0 P where (A, B) = x Ax Bx , Mx = χ̄ax χax , Bx = εabc χax χbx χcx /6, Vx = χ̄x U0 (x)χx+0̂ , βt = d/2Nc2 g2 , βs = (d − 1)/8Nc4 g2 , d = 3 is the spatial dimension, and VM (x, y) and VB (x, y) represent mesonic and baryonic inverse propagators. In SCL, the effective potential for Nc = 3 without baryon effects has been known as,2), 3) 1 (q) (T ) Feff (σ) = aσ σ 2 + Feff (aσ σ + m0 ; T, µ) , 2   (q) Feff (mq ) = −T log Cσ3 − Cσ /2 + cosh (3µ/T ) /4 , (2.4) (2.5) where Cσ = cosh(arcsinh (mq )/T ) and aσ = d/2Nc , and the scalar field σ = −hM i is the chiral order parameter. This effective potential gives the ratio of RµT ∼ 0.33. We have recently demonstrated that the baryonic composite action can be represented in the mean field approximation at zero diquark condensate as,4) (B̄,VB ,B) e h i P (b) −Ns3 Nτ aω ω 2 /2+∆Feff (gω ω) − x [(α2 +γ 2 )M 2 /2+αωM )] ≃e , (2.6) (b) where auxiliary baryon determinant is represented in ∆Feff .4) At equilibrium, the baryon potential field ω is approximately proportional to σ, and the effective potential is found to be (T b) Feff (σ) = 1 (b) (q) bσ σ 2 + Feff (bσ σ) + ∆Feff (gσ σ) . 2 (2.7) Two parameters in this effective potential, bσ and gσ , are related to the decomposition parameters, α and γ. introduced in baryonic composite decomposition. The effective potential with 1/g2 correction was derived by Bilić et al.,3) where (q) V † V term was generated by the derivative of Feff . Since ∆Sg is not in the bilinear form in χ and χ̄, we here bosonize the plaquett contributions and apply the mean T 1.2 SCL (no B) SCL with B SCL w/o B 1 1.5 1.0 T/Tc 0.8 0.6 σ≠0 T σ=0 0.4 0.5 µ 0.5 0.2 Baryon Effects 0 0 0.1 0.2 0.3 µ/Tc 0.4 0.5 5 β=6/g 0.5 µ 2 Fig. 1. The phase diagram in the strong coupling limit (left panel), and its evolution with β = 6/g 2 (right panel). Strong coupling limit/region of lattice QCD 3 field approximation,6)   X βt ϕt X βt 2 βs d 2 Mx Mx+ĵ .(2.8) (Vx − Vx† ) − βs ϕs ϕt + ϕs + ∆SF ≃ Ns3 Nτ 4 4 4 x x,j>0 The auxiliary fields have expectation values of hϕt i = hV † − V i and hϕs i = 2hMx Mx+ĵ i. These correction terms have a similar structure to the SCL effective action (2.2), and they lead to the modifications of the quark mass and effective chemical potential as m e q = σd(1 + 4Nc βs ϕs − βt ϕt cosh µ)/2Nc and µ e = µ − βt ϕt sinh µ. At equilibrium, we can put ϕs = 2σ 2 + O(1/g2 ), and the effective free energy up to O(1/g2 ) without baryon effects is obtained as, (1/g 2 ) Feff = d 2 βt (q) e q ; T, µ e) . σ + 3dβs σ 4 + ϕ2t − Nc βt ϕt cosh µ + Feff (m 4Nc 4 (2.9) As shown in Fig. 1, both of baryons and finite coupling corrections have effects to extend the hadronic phase in the larger µ direction relative to Tc , while each of these is not enough to explain the empirical ratio of RµT . It would be interesting to evaluate both of these effects simultaneously. Chiral RMF with SCL Effective Potential bσ Uσ = tr(MM† ) − aσ log det(MM† ) − cσ σ 2 bσ 2 ∼ σ − 2aσ log σ − cσ σ , (3.1) 2 9 N O Ca Ni E/A(MeV) 8 Zr Sn Si 7 Pb C 6 5 30 4 SΛ(MeV) RMF models are powerful tools in describing nuclear matter and finite nuclei, but we have a so-called chiral collapse problem9) in a naive chiral RMF model based on the φ4 theory; the normal vacuum jumps to an abnormal one below the normal nuclear density. The effective potential in SCL-LQCD gives us a hint to solve this problem. In a zero temperature treatment of SCL-LQCD,1), 4) the effective potential is found to have the form, Feff = bσ σ 2 /2 − Nc log σ, and the divergent behavior at σ → 0 helps to avoid the chiral collapse. We have recently developed a chiral SU(2) RMF model7) with logarithmic sigma potential, Chiral SU(3) exp. Λ s p d f 20 10 Chiral SU(3) exp. 0 1000 500 Σ Shift (eV) §3. 10->9 4->3 100 50 5->4 Chiral SU(3)(SR) Chiral SU(3)(WR) Exp. 10 0 5 10 15 20 A2/3 25 30 35 40 Fig. 2. Nuclear binding energies (top), single Λ hypernuclear separation energies (middle), and atomic shift of Σ − atom (bottom) in the present chiral SU(3) RMF. where M denotes the meson matrix. In this RMF, we can well describe symmetric nuclear matter equation of state (EOS) and bulk properties of finite nuclei.7) In a chiral SU(3) RMF model,8) we include the determinant interaction (det M+det M† ) 4 A. Ohnishi, N. Kawamoto, K. Miura, K. Tsubakihara, H. Maekawa simulating UA (1) anomaly. After fitting binding energies and charge rms radii of normal nuclei, we find that the symmetric matter EOS becomes softer than that in the chiral SU(2) RMF due to the scalar meson with hidden strangeness, ζ = s̄s, which couples with σ through the determinant interaction. In this chiral SU(3) RMF, we can find hyperon-meson coupling constants which fits existing data of Λ separation energies in single hypernuclei, ΛΛ bond energy (∆BΛΛ ) in 6ΛΛ He, and atomic shifts of Σ − atoms, as shown in Fig. 2. While the above sigma potential has divergence at σ → 0, the effective potential derived in the finite T treatments have a linear term for small values of σ at T → 0, aσ 2 (T ) Feff → σ − Nc arcsinh (aσ σ) , (3.2) 2 which is enough to stabilize the normal vacuum. We may conclude that gluons play decisive roles to avoid the chiral collapse as pointed out in Ref. 10). §4. Summary In this paper, we have investigated the phase diagram of quark matter and nuclear properties based on the strong coupling expansion of lattice QCD. In the first part, we find that baryons and finite coupling corrections have favorable effects to extend the hadron phase to a larger µ direction with respect to Tc . In the second part, we have shown that we can well describe normal and hypernuclear properties with a logarithmic potential. For the understanding of dense matter, it is important to respect both of chiral symmetry and strangeness degrees of freedom. In addition, the present work may be suggesting the importance of implicit role of gluons, which generate the effective potential of hadrons. This work is supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research under the grant numbers, 13135201, 15540243, and 1707005. References 1) N. Kawamoto and J. Smit, Nucl. Phys. B 192 (1981), 100. 2) P. H. Damgaard, N. 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