Novel Effects of Electromagnetic Interaction on the Correlation of Nucleons in Nuclear Matter

Physics Letters B 608 (2005) 39–46 www.elsevier.com/locate/physletb

Novel effects of electromagnetic interaction on the correlation of nucleons in nuclear matter

Ji-Sheng Chen ∗, Jia-Rong Li, Meng Jin

Institute of Particle Physics and Physics Department, Hua-Zhong Normal University, Wuhan 430079, People’s Republic of China Received 19 April 2004; received in revised form 27 September 2004; accepted 27 December 2004 Available online 5 January 2005 Editor: W. Haxton

Abstract The electromagnetic (EM) interactions between charged protons on the correlations of nucleons are discussed by introducing the Anderson–Higgs mechanism of broken U(1) EM symmetry into the relativistic nuclear theory with a parametric photon mass. The non-saturating Coulomb force contribution is emphasized on the equation of state of nuclear matter with charge 1 symmetry breaking (CSB) at finite temperature and the breached S0 pairing correlations of proton–proton and neutron–neutron. The universal properties given by an order parameter field with a non-zero vacuum expectation value (VEV) nearby phase transition are explored within the mean field theory (MFT) level. This mechanism can be extended to the charged or charge neutralized strongly coupling multi-components system for the discussion of binding or pairing issues.  2004 Elsevier B.V. All rights reserved.

PACS: 21.30.Fe; 21.10.Sf; 11.30.Cp

Understanding the properties of nuclear matter un- problem in contemporary physics [1,2]. The dis- der both normal and extreme conditions is of great cussion about the property of nuclear ground state- importance in relativistic heavy ion collisions and ex- binding energy and pairing correlations at low temper- plaining the appearance of compact objects such as ature is substantial. the neutron stars and neutron-rich matter or nuclei. The in-medium behavior associated with the many- The determination of the properties of nuclear mat- body characteristic is the key, while a non-perturbative ter as functions of density/temperature, the ratio of approach is crucial. The theoretical difﬁculty of mak- protons to neutrons, and the pairing correlations— ing low energy calculation directly with the fundamen- superﬂuidity or superconductivity is a fundamental tal quantum chromodynamics (QCD) makes effective theories still desirable. As accepted widely, the relativistic nuclear theory can successfully describe the * Corresponding author. saturation at normal nuclear density and the spin– E-mail address: [email protected] (J.-S. Chen). orbit splitting [3,4]. The further developments [5] of

σ –ω theory of quantum hadrodynamics model (QHD) The pure neutron matter cannot exist in nature, and make it possible to determine the model parameters the realistic nuclear matter is subject to the long range analytically from a specified set of zero-temperature EM interaction. The changes of symmetry proper- nuclear properties and allow us to study the hot nu- ties associated with possible phase transition realized clear properties and study variations of these results on some conditions attract physicists very much. In to nuclear compressibility or pairing correlation, and nuclear physics, charge symmetry breaking explored even the symmetry energy coefficient according to by the quite different empirical negative scattering baryon density [6], which are not well known. Fur- lengths aNN(P) and aPP is a fundamental fact [17] and thermore, the in-medium hadronic property has at- there are existed works to address its theoretical ori- tracted much attention with this kind of models and gin [18]. Coulomb correlation effects are a fundamen- there are many existed works although the relation be- tal problem in nuclear physics and play an important tween QCD and QHD has not been well established role for the property of nuclear matter [19,20],which [7,8]. In recent years, with the refinement of nuclear may lead to rich phase structures in the low temper- theory study, the σ –ω theory has been placed in the ature occasion. For example, in Ref. [21] the influ- context of effective theory and it is argued that the ence of the non-saturating Coulomb interaction is re- vacuum physics has been explored in part by this cently incorporated in the multi-canonical formalism kind of models [9,10]. In physics, with the obvious attempting to explain the reported experimental sig- non-vanishing fermion nucleon mass in relevant La- natures of thermodynamic anomalies and the possible grangian, the hidden chiral symmetry is explicitly bro- liquid–gas (LG) phase transitions of charged atomic ken. clusters and nuclei [22]. One may naturally worry The theoretical S-wave pairing correlation issue is about the important role of the Coulomb repulsion 1 a long-standing problem. The fundamental S0 pair- force on the properties of charged system and the ther- ing in infinite nuclear matter within the frame of rel- modynamics of charged/neutral nuclear matter to be ativistic nuclear field theory was first discussed by reflected by relativistic nuclear theory and correspond- Kucharek and Ring [11], and it was found that the ing approaches. Within the models based on σ –ω field gaps are always larger for three times than the non- theory and usual adopted approaches such as MFT relativistic results [12,13]. Especially, the very un- or relativistic Hartree approximation (RHA), one can 1 comfortable non-zero gaps of S0 pairing correla- suppose the similar interactions between PP and NP tion at zero baryon density obtained with frozen me- or NNs, with the weak EM interaction being neglected son propagators in relativistic field theory, as recently compared to the residual strong interaction between pointed out by us [14], remind us that the realis- nucleons. Theoretically, the direct (Hartree) Coulomb tic nuclear ground state with MFT approach might contribution of charged protons to the EOS cannot be not be EM empty. On the other hand, the well es- included due to the Furry theorem’s limit. Although tablished low temperature superconductivity theory the exchange (Fock) contribution can be included in tells us that it would be very interesting to discuss principle from the point of view of field theory, the in- the broken local EM symmetry effects on the proper- volved calculation and radioactive corrections caused ties of the nucleons system. Although the in-medium by relevant infrared singularity of photon propagator nucleon–nucleon interaction potential induced by po- still remain to be done even in the relatively simpler larization can give a significantly improved descrip- zero-temperature occasion in nuclear physics. If one tion for EOS and superfluidity [8,14,15], the pairing asks what the difference between PP and NP or NN difference of PP (proton–proton) from NP (neutron– pairing correlation is, the original version of QHD proton) or NN (neutron–neutron) has been discarded. with MFT or RHA approaches cannot tell us anything. Other approaches also recently found that polariza- Although one can expect that the isospin breaking tion effects suppress the S-wave gaps by a factor of coupling terms such as ρNN, etc., might reflect the 1 3–4 [16]. The numerical magnitude of S0 gaps is not Coulomb repulsion contribution on the thermodynam- sensitive to a special parameters set and integral mo- ics of isospin asymmetric system to some extent, the mentum cutoff when the polarization effect is taken pairing differences between PP and NN, NP exist even into account [14]. for symmetric nuclear matter incorporated with the J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46 41 quite different empirical negative scattering lengths by the surrounding such as electrons to maintain the aNN(P) and aPP. How to incorporate the important role stability for compact object through β-equilibrium). of EM interaction with CSB on the thermodynamics Also, the quartic–cubic terms of σ non-linear interac- of charged/neutral system or the property of nuclear tion potential U(σ) = bσ3 + cσ 4 with the additional ground state on a microscopic level (continuum field phenomenologically determined parameters b and c theory) remains an intriguing task even in an oversim- have not been obviously preferred in order to discuss plified way (MFT or RHA) but with thermodynamics in a more general way although a specific assumption self-consistency. in U(σ)can give a reasonable bulk compressibility for In this Letter, we propose a systematic way to per- nuclear matter. form the link between the bulk and pairing corre- The mean field approximation can be used to dis- lation many-body properties of charged/neutral two- cuss the thermodynamics of charged nuclear matter, components nucleon systems through a relativistic nu- from which the effective potential is derived in terms clear field theory involving the interaction of Dirac nu- of finite temperature field theory [3,4,25] cleons with massive photons as well as the well-known scalar/vector mesons. Inspired by the continuum field Ω = 1 2 2 − 1 2 2 − 1 2 2 − 1 2 2 mσ φ0 mωω0 mρρ03 mγ A0 theory of phase transition and based on QHD-II, the V 2 2 2 2 constructed phenomenological Proca-like Lagrangian 2 3 −β(E∗−µ∗) − T d k ln 1 + e i i through Anderson–Higgs mechanism is [3,4,23,24] (2π)3 i −β(E∗+µ∗) ¯ µ µ + ln 1 + e i i , (2) L = ψ iγµ∂ − M − gσ σ − gωγµω = + where i P , N represents the index of proton (P) and 1 µ 1 τ3 µ − gργµτ ·ρ − eγµ A ψ neutron (N), respectively, and V is the volume of the 2 2 system. With the thermodynamics relation + 1 µ − 1 2 2 − 1 µν ∂µσ∂ σ mσ σ HµνH 1 ∂(βΩ) 2 2 4 = + µ ρ , V ∂β i i 1 2 µ 1 µν 1 2 µ i + m ωµω − Rµν · R + m ρµ ·ρ 2 ω 4 2 ρ one can obtain the energy density − 1 µν + 1 2 µ Fµν F mγ AµA 4 2 m2 g2 + L = σ (M − M∗)2 + ω ρ2 δ Higgs&counterterm, (1) 2 2 B 2gσ 2mω µ µ µ where σ , ω , ρ and A are the scalar–isoscalar, 2 2 gρ 2 e 2 vector–isoscalar, vector–isovector meson fields, EM + (ρP − ρN ) + ρ 8m2 2m2 P field with the field stresses ρ γ 2 3 ∗ ∗ ∗ = − + d k E ni (µ ,T)+¯ni (µ ,T) Hµν ∂µων ∂ν ωµ, (2π)3 i i i i Rµν = ∂µρν − ∂ν ρµ − gρ(ρµ ×ρν ), (3) and pressure p =−Ω/V. The baryon density is Fµν = ∂µAν − ∂ν Aµ ¯ for ω, ρ and Aµ’s, respectively. The M, mσ , mω, mρ ρB =ψψB = ρi, and m are the nucleon, meson and photon masses, γ i while gσ , gω, gρ and e are the coupling constants for 2 3 ρi = d k (ni −¯ni). (4) corresponding Yukawa-like effective interaction, re- (2π)3 spectively. ∗ ¯ ∗ Here the Lagrangian with CSB does not respect the In above expressions, ni(µi ,T), ni (µi ,T) are the ∗ = local UEM(1) gauge symmetry which is broken by the √distribution functions for (anti-)particles with Ei ground state with non-zero local electric charge of pro- k2 + M∗ 2. The effective nucleon mass M∗,chem- ∗ tons (although the system can be globally neutralized ical potentials µP(N) are introduced by the tadpole 42 J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46 diagrams of the sigma, omega, rho mesons and photon self-energies, respectively: 2 ∗ ∗ 2 gσ 3 M M = M − d k (ni +¯ni); (5) (2π)3 m2 E∗ σ i g2 1 g2 e2 µ∗ = µ − ω ρ − ρ (ρ − ρ ) − ρ , P P 2 B 2 P N 2 P mω 4 mρ mγ 2 g2 ∗ = − gω + 1 ρ − µN µN 2 ρB 2 (ρP ρN ), (6) mω 4 mρ where µP(N) is the proton (neutron) chemical potential. The photon mass mγ appears as a free parameter which is closely related to the Coulomb energy (reflecting the binding energy contributed by adding aprotonto or removing a neutron from the system) and correspondingly to Coulomb compression modulus KC . It is worthy noting that the Coulomb energy can be discussed by the conventional many-body approaches such as the Thomas–Fermi theory with vari- ational principle [26]. Within relativistic MFT, the KC has been analyzed in the literature such as in Ref. [27] with the scaling model [28] phenomenologically. The Fig. 1. For charged nuclear matter without considering the charge bulk compression modulus K and KC are defined by neutral condition with the set (a) of Table 1: (a) Pressure ver- 2 sus rescaled density ρB /ρ with (solid) and without (dashed) the 2 ∂ eb 0 = K K 9ρ0 2 , Coulomb repulsion interaction, (b) C versus the order parameter ∂ρ = B ρB ρ0 mγ describing to what extent the EM symmetry is broken. 3α 9K KC =− + 8 . (7) 5R K 0 bility in the high baryon density region caused by a Here, ρB , ρ0, eb are the baryon density, the normal negative parameters set of b and c in the non-linear baryon density and the binding energy per nucleon self-interaction term U(σ) (for obtaining a reason- with able compressibility modulus of bulk EOS). One can 3 1/3 d3e estimate the parameter mγ is about 20–30 MeV for = = 3 b ∼ R0 ,K3ρ0 3 . a reasonable critical temperature Tc 16 MeV ac- 4πρ0 dρ = B ρB ρ0 cessible in heavy ion collisions. In Fig. 1,wegive The repulsive Coulomb force will modify the EOS sig- the curves of pressure versus baryon density and the nificantly for the realistic charged system produced Coulomb compression modulus KC according to the in heavy ion collisions. Especially, it can make the interaction strength characterized by mγ with frozen critical temperature Tc of the LG phase transition de- parameters gσ and gω which are determined by fit- 0 =− creased to a smaller value. With careful numerical ting the binding energy eb 15.75 MeV and the study, it is found that the softness of bulk EOS (char- bulk symmetry energy coefficient asym = 35 MeV (for acterized by K) is not sensitive to the Coulomb inter- symmetric nuclear matter at the empirical saturation −3 action but the critical temperature Tc as well as KC density ρ0 = 0.1484 fm with T = 0) [4]. The qual- is very sensitive to this repulsive force. The additional itative Coulomb effect on the deformation of phase Coulomb energy term in pressure and energy density space distribution functions resulting from Eq. (4) and Eq. (3) contributes to removing the theoretical insta- Eq. (6) can be reflected by the proton fraction ratio: J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46 43

Table 1 = = = = 2 = 2 2 2 The parameters are with M 939, mρ 770, mω 783, mσ 520 MeV(s) and mγ is in (MeV). Ci gi M /mi ∗ 2 2 2 2 2 M Set g g g (C ) mγ (C ) σ ω ρ ρ γ M ρ0 MFT a 91.64 191.05 6.91 (10.28) 30.44 (87.27) 0.540 b 0 28.79 (97.55) c65.58 (97.55) ∞ or e = 0(0)

RHA d 69.98 102.76 6.91 (10.28) 26.636 (113.96) 0.731 e 0 25.51 (124.24) f83.54 (124.24) ∞ or e = 0(0)

YP = ρP /ρB . It is found that this ratio changes sig- topic can contribute to giving a solid limit for fixing nificantly according to temperature T and total baryon mγ and gρ . The large tensor and spin–orbit forces are density ρB . also crucial for understanding finite nuclei and neu- Therefore, the electric repulsive force plays an tron star structure which can be explored through the isospin violating role for the many-body property. study for mirror-nuclei. Let us mention that the hith- Indeed, there is some kind competition between the erto overlooked but important EM interaction role on ρ and photon’s isospin breaking effect on the phase the spin–orbit splitting for some mirror-nuclei is re- space distribution function deformation. The Coulomb cently found in Ref. [30]. force makes the proton fraction decreased while the To study its effect on the electric neutral nuclear ρ meson plays a weak inverse role. Furthermore, one matter such as compact proto-neutron star would also can readily derive the symmetry energy coefficient for- be interesting. For the simplest NP+e+νe system sta- mula at T = 0, bilized through β-equilibrium, charge neutrality condition makes the proton fraction ratio very small. It 1 ∂2(/ρ) a = is found that the Coulomb force does not modify this sym 2 2 ∂t t=0 picture as indicated by Fig. 2. This is consistent with

k3 g2 e2 k2 above result that the Coulomb interaction does not = f ρ + + f 2 2 2 , change the softness of bulk EOS significantly. 12π mρ mγ 2 + ∗ 2 6 kf M Correlations not only do manifest themselves in − the bulk properties but also modify the quasi-particle = ρN ρP t . (8) properties of nucleons in a substantial way. Concep- ρB tually, the PP and NN(P) pairing correlations should In fact, if without taking into account the repulsive be quite different from each other. The former has Coulomb contribution, one must introduce a very large additional superconductivity contribution due to the coupling constant gρ to approach the empirical sym- electric charge of protons in addition to the attractive metric coefficient asym which is very far from the residual strong interaction compared with the scenario empirical coupling constant gρNN extracted experi- of NN correlation. For the fundamental 1S pairing, = 0 mentally. From Eq. (8),iftakinggρ 2.63 [7] and the energy gap equation of nucleon–nucleon pairing in = asym 35 MeV, the free parameter mγ can be fixed the frame of relativistic nuclear theory can be reduced accordingly. With close study, the numerical magni- ∗ to [11,13,14] tude of mγ is more sensitive to M (and hence the softness of bulk EOS) than to g . The relevant pa- ρ =− 1 ¯ ∆(k) 2 rameters are listed in Table 1 corresponding to the ∆(p) vpp (p, k) k dk, 8π2 (k)2 + ∆2(k) L2 set of Ref. [4] except of taking gρ = 2.63. Fur- ther exact fitting to finite nuclei data such as charge (9) density radii distribution, etc. [4,29], and addressing and the coupled effective mass gap equation has been the spin–orbit splitting issue with the mirror-symmetry neglected here for brevity. 44 J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46

Fig. 2. For electric neutral nuclear matter stabilized through β-equilibrium: (a) Pressure versus rescaled density. (b) Proton frac- Fig. 3. (a) Pairing gap ∆f at the Fermi surface versus Fermi tion ratio YP . Line-styles are similar to Fig. 1. momentum kf . (b) Gap function ∆(k) versus momentum k for fixed Fermi momentum. The solid line corresponds to the result of proton–proton pairing correlation and dashed line to that of neu- In Eq. (9), the asymmetrized matrix elements tron–neutron. v¯pp (p, k) is obtained through the integration of v(¯ p, k) over the angle θ between the three-momentums p and between PP and NN pairing correlations reflects that ¯ k with v(p, k) being the particle–particle interaction virtual photons proceeding in the space-like momen- potential tum transfer regime carry a unique information on the ∗2 EM properties of nucleon interaction responsible for ¯ =∓ M v(p, k) ∗ ∗ the nucleon structure. As indicated by Fig. 3, the long 2E (k)E (p) range but screened Coulomb interaction affects the + + Tr[Λ+(k)Γ+(p)γ 0T Γ T γ 0] correlation function of proton–proton pairing signifi- × , − 2 + 2 cantly, especially in the low momentum regime. (k p) mD ∗ The physical reason for the parametric descrip- = k/+M where Λ+(k) 2M∗ is the projection operator of the tion of the EM interaction in this approach is that positive energy solution and T = iγ1γ 3 is the time at first one can note the existence of locally charged reversal operator. The Γ is the corresponding inter- system/cluster, i.e., the electro-magnetic field con- ¯ µ action vertex of σ/ω, ρ(γ) with nucleons while mγ densation ∼ψP γ ψP (corresponding to the spon- is the photon mass. This static electric contribution taneously breaking of local gauge symmetry while to the gap has been indicated in Fig. 3 as a curve of the gauge field obtains mass) in the low energy gap versus density with a integral momentum cutoff scale although the stable system should be globally −1 Λk = 3.6fm and the set (a) in Table 1 to numer- neutral with surrounding such as electrons through ically solve the integral gap equation. The difference β-equilibrium. This is very much similar to the chi- J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46 45 ral condensation ∼ψψ¯ at low energy scale. Sec- the strength. Especially, the repulsive Coulomb force ond, from the point of view of continuum field the- makes the critical temperature lower than the existed ory with symmetry changes, the physics background theoretical anticipation and contributes to interpret- of well-known low temperature LG phase transition ing the accessible experimental results. Furthermore, still remains to be explored. Especially, how to re- the breached PP and NN pairing correlation strengths flect the CSB characteristic in relevant effective theory open a new window for the study of nuclear matter and approaches remains to be performed. Third, in EM property/nuclei structure and would lead to rich the multi-components Fermi/Bose systems the CSB physical phenomena. From the view of point of many- would lead to more rich phenomena, e.g., compared body physics, the low-temperature LG phase transition with the metal electric superconductivity occasion (the found in heavy ion collisions and the different cor- ions fixed as lattice). relation strengths for PP and NN(P) bound-state can From the point of view of Maxwell QED, be- be seen as the fingerprint of broken EM symmetry cause photons are massless, photon-mediated inter- within MFT to some extent. Our discussion based on actions are long range in contrast with a point- assuming the spontaneously broken EM gauge sym- like meson–nucleon interaction in the existed QHD- metry highlights that the U(1) electric charge sym- like Lagrangian. The long range nature of photon- metry violating effects should be taken into account exchange manifests itself in the infrared singular be- simultaneously with the SU(2) isospin breaking ef- havior of the photon propagator. This characteristic fects played by such as ρNN coupling. The weak cou- enhances the contribution of very soft, collinear pho- pling interaction is mixed with other stronger ones and tons to the correlation energy for the EOS or the pair- plays an important role for the many-body effects. Es- ing problem by noting that this divergence should be pecially, the overlooked EM interaction contribution avoided by the resummation approach as done in QCD on the many-body properties such as thermodynam- or QED. Essentially, different from the QCD occasion ics, binding, pairing mechanism, etc., of nucleons in (with the magnetic mass cutoff due to the non-Abelian nuclear matter should be carefully considered from the self-interaction of gluons) [31], there is no magnetic point of view of continuum field theory. screening in QED, which makes it very involved to discuss the superconductive behaviors in strong magnetic field occasion such as in compact star environ- Acknowledgements ment with conventional QHD-like Lagrangian. This approach makes it possible to further study Meissner Ji-sheng Chen acknowledges the beneficial discus- as well as Debye screening effects in such as astro- sions with Prof. Hong-an Peng (Peking University). physics [32]. The premise of this approach as a non- This work was supported by NSFC under grant Nos. local effective theory nearby a phase transition with 10175026, 90303007. CSB through one-meson (photon) exchange picture would be very powerful in addressing nuclear matter (either symmetric or asymmetric) many-body properties and even those of finite nuclei. References In summary, the long range non-saturating Cou- lomb interaction plays an important role in the prop- [1] P. Daniellewicz, R. Lacey, W.G. Lynch, Science 298 (2002) erty of charged/neutral nuclear matter, which can be 1592; J.M. Lattimer, M. Prakash, Science 304 (2004) 536. incorporated simultaneously with the residual strong [2] D.J. Dean, M. Hjorth-Jensen, Rev. Mod. Phys. 75 (2003) 607. interaction within MFT of relativistic nuclear theory [3] J.D. Walecka, Ann. Phys. (N.Y.) 83 (1974) 491; through an effective Proca-like Lagrangian. The defor- B.D. Serot, J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1. mation of phase-space distribution function of nucle- [4] B.D. Serot, J.D. Walecka, Int. J. Mod. Phys. E 6 (1997) 515. ons attributed to the static electric interaction can man- [5] J. Boguta, A.R. Bodmer, Nucl. Phys. A 292 (1977) 413; J. Zimanyi, S.A. Moszkowski, Phys. Rev. C 42 (1990) 1416. ifest itself on the thermodynamics or the property of [6] B.-A. Li, Phys. Rev. Lett. 85 (2000) 4221. ground state of charged nuclear matter and the quasi- [7] For example, H. Shiomi, T. Hatsuda, Phys. Lett. B 334 (1994) particle spectrum while the photon mass mγ controls 281; 46 J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46

H.-C. Jean, J. Piekarewicz, A.G. Williams, Phys. Rev. C 49 [17] For example, R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, Phys. (1994) 1981; Rev. C 51 (1995) 38; J.-S. Chen, J.-R. Li, R.-F. Zhuang, J. High Energy Phys. 11 R. Machleidt, Phys. Rev. C 63 (2001) 024001; (2002) 014, nucl-th/0202026; T. Duguet, Phys. Rev. C 69 (2004) 054317. J.-S. Chen, J.-R. Li, R.-F. Zhuang, Phys. Rev. C 67 (2003) [18] S. Gardner, C.J. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 75 068202. (1995) 2462, and references therein. [8] J.-S. Chen, P.-F. Zhuang, J.-R. Li, Phys. Rev. C 68 (2003) [19] A. Bulgac, V.R. Shaginyan, Phys. Lett. B 469 (1999) 1; 045209. M. Bender, P.H. Heenen, P.G. Reinhard, Rev. Mod. Phys. 75 [9] G.E. Brown, M. Rho, Phys. Rev. Lett. 66 (1991) 2720; (2003) 121. R. Rapp, R. Machleidt, J.W. Durso, G.E. Brown, Phys. Rev. [20] T. Maruyama, et al., nucl-th/0402002, and references therein. Lett. 82 (1999) 1827; [21] F. Gulminelli, et al., Phys. Rev. Lett. 91 (2003) 202701. G.E. Brown, M. Rho, Phys. Rep. 363 (2002) 85; [22] P.J. Siemens, Nature (London) 305 (1983) 410; C. Song, Phys. Rev. Lett. 347 (2001) 289. M. Schmidt, et al., Phys. Rev. Lett. 86 (2001) 1191; [10] G. Gelmini, B. Ritzi, Phys. Lett. B 357 (1995) 431; F. Gobet, et al., Phys. Rev. Lett. 89 (2002) 183403; G.E. Brown, M. Rho, Nucl. Phys. A 596 (1996) 503. S. Siem, et al., Phys. Rev. C 65 (2002) 044318. [11] H. Kucharek, P. Ring, Z. Phys. A 339 (1991) 23. [23] M.E. Peskin, D.V. Scroeder, An Introduction to Quantum Field [12] J. Decharge, D. Gogny, Phys. Rev. C 21 (1980) 1568. Theory, Weber State University, Ogden, UT, 1995. [13] For example, M. Matsuzaki, Phys. Rev. C 58 (1998) 3407; [24] A. Jenkins, Phys. Rev. D 69 (2004) 105007, and references M. Serra, A. Rummel, P. Ring, Phys. Rev. C 65 (2001) 014304; therein. B.V. Carlson, T. Frederico, F.B. Guimarães, Phys. Rev. C 56 [25] J.I. Kapusta, Finite Temperature Field Theory, Cambridge (1997) 3097; Univ. Press, Cambridge, 1989. B.F. Haas, T. Frederico, B.V. Carlson, F.B. Guimarães, Nucl. [26] For example, K.-S. Cheng, C.-C. Yao, Z.-G. Dai, Phys. Rev. 55 Phys. A 728 (2003) 379. (1997) 2092; [14] J.-S. Chen, P.-F. Zhuang, J.-R. Li, Phys. Lett. B 585 (2004) 85. H. Shen, Phys. Rev. C 65 (2002) 035802. [15] X.-D. Ji, Phys. Lett. B 208 (1988) 19; [27] H. Kouno, et al., Phys. Rev. C 53 (1996) 2542. A. Bhattacharyya, S.K. Ghosh, S.C. Phatak, Phys. Rev. C 60 [28] J.-P. Blaizot, Phys. Rep. 64 (1980) 171. (1999) 044903. [29] C.J. Horowitz, J. Piekarewicz, Phys. Rev. Lett. 86 (2001) 5647. [16] For example, A. Schwenk, B. Friman, Phys. Rev. Lett. 92 [30] J. Ekman, et al., Phys. Rev. Lett. 92 (2004) 132502. (2004) 082501; [31] For example, D.H. Rischke, Phys. Rev. D 62 (2000) 054017. A. Schwenk, B. Friman, G.E. Brown, Nucl. Phys. A 713 (2003) [32] K.B.W. Buckley, M.A. Metlitski, A.R. Zhitnitsky, Phys. Rev. 191; Lett. 92 (2004) 151102; A. Schwenk, B. Friman, G.E. Brown, Nucl. Phys. A 703 (2002) K.B.W. Buckley, M.A. Metlitski, A.R. Zhitnitsky, Phys. Rev. 745. C 69 (2004) 055803.