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

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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 rela- tivistic 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 0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.12.074 40 J.-S. Chen et al. / Physics Letters B 608 (2005) 39–46 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.
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