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Chiral Symmetry and Scalar Meson in Hadron and Nuclear Physics

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Chiral Symmetry and Scalar Meson in Hadron and Nuclear Physics RYUTHP 95-1 February 1995 Chiral Symmetry and Scalar Meson in Hadron and Nuclear Physics 1 Teiji Kunihiro Faculty of Science and Technology, Ryukoku University, Seta, Otsu-city 520-21, Japan Abstract After giving a short introduction to the Nambu-Jona-Lasinio model with an anomaly term, we show the importance of the scalar-scalar correlation in the low-energy hadron dynamics, which correlation may be summarized by a scalar-isoscalar meson, the sigma meson. The discussion is based on the chiral quark model with the sigma-meson de- grees of freedom. Possible experiments are proposed to produce the elusive meson in a nucleus and detect it. In relation to a precursory soft mode for the chiral transition, the reason is clarified why the dynamic properties of the superconductor may be described by the diffusive time-dependent Ginzburg-Landau (TDGL) equation. We indicate the arXiv:hep-ph/9502305v1 15 Feb 1995 chiral symmetry plays a significant role also in nuclei; one may say that the stability of nuclei is due to the chiral symmetry of QCD. 1Invited talk presented at YITP workshop on “From Hadronic Matter to Quark Matter: Evolution of Hadronic Matter”. Originally, the talk was entitled ”Hadron Properties and Chiral Transition in an Effective Theory” by the organizers of the workshop. To be published in Prog. Theor. Phys. Supplement (Kyoto). 1 1 Introduction The salient features of low-energy hadron physics or QCD may be summarized as follows; (i) Confinement of the colored quarks and gluons, (ii) the dynamical breaking of chiral symmetry (DBCS), (iii) UA(1) anomaly, (iv) Explicit SUV (3) breaking, (v) Success of the constituent quark model, (vi) OZI rule and its violation in mesons and baryons. Hatsuda and the present author [1] showed that a semi-phenomenological but unified description of the above facts is possible, emphasizing a dominant role of chiral symmetry in low-energy hadron dynamics. The description is based on the SU(3) Nambu-Jona-Laisnio (NJL) model[2]. The model embodies three basic ingredients of QCD, i.e., the dynamical breaking of chiral symmetry (DBCS), UA(1) anomaly and the explicit symmetry breaking due to the current quark masses. Various empirical aspects of QCD were shown to be realized through interplay among the three ingredients. It was emphasized that the constituent quark model and the chiral symmetry is reconciled in a chiral quark model. It was also shown that the chiral quark model can account for most of the empirical facts on baryons as well as the low-lying mesons. Furthermore, the NJL model allows us to study the change of hadron properties in hot/dense medium in a self-consistent way. In this report, we shall first give a short introduction of the NJL model with the anomaly term in section 2, which is the basis for the proceeding discussions. Then we pick up and rearrange some topics dealt in the above review article to clarify significance of the sigma meson in the low-energy hadron dynamics. The sigma meson is a scalar and iso-scalar meson and the chiral partner of the pion; see section 3. We show that a chiral quark model incorporating the collectiveness in the scalar channel summarized by the sigma meson nicely accounts for some properties of lowlying baryons, such as π-N Σ term[3]. We also mention that the convergence radius of the chiral perturbation theory is related with the sigma meson mass. In section 4, possible experiments are proposed to detect the elusive meson in the laboratory. Next, we move to the problem of the fluctuation effects in hot QCD. This is also a topics dealt in Ref. [1], where it is shown that a precursory soft mode exist in the high temperature phase for the chiral transition in QCD. We shall make a comment on the nature of the corresponding mode in the weak-coupling supercondoctor as described by the BCS model[4]: It will be clarified why the dynamic properties of the superconductor may be usually described by the diffusive time-dependent Ginzburg-Landau (TDGL) model; see section 5. Finally, we indicate that the chiral symmetry plays a significant role in nuclei as well as in one hadron systems. Notice that nuclei are only stable bound systems in the hadron world. One may say that this stability is due to the chiral symmetry of QCD; see section 6. 2 Brief summary of the SU(3) NJL model with an anoamly term Our model Lagrangian is the generalized Nambu-Jona-Lasinio (NJL) model with the anomaly term[5, 6, 7]; 8 gS 2 2 =qiγ ¯ ∂q + [(¯qλaq) + (¯qiλaγ5q) ] q¯mq + gD [det¯qi(1 γ5)qj + h.c.], L · aX=0 2 − − 2 + + + (2.1) ≡ L0 LSB LS LD a where the quark field qi has three colors (Nc = 3) and three flavors (Nf = 3), λ 2 (a=0 8) are the Gell-Mann matrices with λ0= 1. Here 0 + S NJL is the ∼ q 3 L L ≡ L U(3) generalization of the NJL model and has manifest flavor-U (3) U (3) invariance. L ⊗ R is the explicit SU (3) breaking part with m (i=u, d, s) being the current quark LSB V i mass. Finally, in (2.1) denotes a term which has SU (3) SU (3) invariance but LD L ⊗ R breaks the UA(1) symmetry[8, 9]; this term is a reflection of the axial anomaly in QCD. While does not cause the flavor mixing, the anomaly term does; with the dynamical LS breaking of chiral symmetry, it induces effective 4-fermion vertices such as dd¯ (¯uu)(¯ss) h i and dd¯ (¯uiγ u)(¯siγ s), where the former (latter) gives rise to a flavor mixing in the −h i 5 5 scalar (pseudo-scalar) channels. For notational convenience, we here introduce the bosonic variables Φ =q ¯ (1 ij j − γ )q : Note thatq ¯(1 γ )λ q = Tr[λ Φ]. The fact that represents the U (1) 5 i − 5 a a LD A anomaly can be seen in the anomalous divergence of the flavor singlet axial current µ µ ¯ µ µ J5 =uγ ¯ γ5u + dγ γ5d +sγ ¯ γ5s, ∂ J µ = 4N g Im(detΦ) + 2iqmγ¯ q, (2.2) µ 5 − f D 5 which is to be compared with the usual anomaly equation written in terms of the µ 2 2 a ˜µν topological charge density of the gluon field, ∂µJ5 = 2Nf g /32π Fµν Fa + 2iqmγ¯ 5q. Thus one may say that the determinantal 6-fermion operator 2gD Im(detΦ) simulates 2 − g a ˜µν the effect of the gluon operator 32π2 Fµν Fa [10] in the quark sector. For further comments on the model lagrangian and the details of the following discussions, see the review paper [1] or the original papers cited in the beginning of this section. In the self-consistent mean field theory, the Lagrangian can be rewritten as follows; = + , (2.3) L LMF A Lres where =q ¯(iγ ∂ M)q g Tr(φ†φ) 2g (detφ + c.c.), LMF A · − − S − D = g : Tr(Φ†Φ) : Lres S 1 1 +g : [Tr(φΦ2) Tr(φΦ)TrΦ TrΦ2Trφ + Trφ(TrΦ)2 +h.c.] : D − − 2 2 +gD : (detΦ+h.c.) : . (2.4) Here the Fock terms are omitted, the normal ordering is taken with respect to the Fock vacuum of and φ is a diagonal 3 3 matrix defined in terms of the quark LMFA × condensates as φ = Φ diag.( uu¯ , dd¯ , ss¯ ) (2.5) h i0 ≡ h i h i h i The “constituent quark mass matrix” M = diag.(Mu, Md, Ms) is given in terms of the condensates M = m 2g uu¯ 2g dd¯ ss¯ , (2.6) u u − sh i− D h ih i Md and Ms with the subscripts u, d and s being replaced in a cyclic way. The quark condensates are in turn given with Mi; 4 Λ d p 1 Nc 2 Mi uu¯ = iNcTr 4 = 2 p dp . (2.7) h i − Z (2π) γ p M − π Z0 M 2 + p2 · − q i 3 Here the three momentum cut-off Λ is introduced, and dd¯ ( ss¯ ) is obtained by the h i h i replacement M M (M M ). u→ d u→ s The q-¯q collective excitations above the condensed vacuum are formed due to the residual interaction in (2.4). The 4-fermion part of can be decomposed into Lres Lres physical channels; 1,2 4,5 6,7 0,3,8 (4) 1 2 S = [(G +G ± +G 0 ) : S :+ : S G S :] Lres 2 δ κ κ a a ab b Xa Xa Xa Xa,b 1,2 4,5 6,7 0,3,8 1 2 P + [(G +G ± +G 0 ) : P :+ : P G P :] , (2.8) 2 π K K a a ab b Xa Xa Xa Xa,b where we have introduced the composite operators Sa =qλ ¯ aq and Pa =qiγ ¯ 5λaq. (4) The coupling constants (G’s) in res are summarized in Table 2.1 of Ref.[1]. The suffix L of the coupling constants shows the relevant channel where the residual interaction is active. We note that not only but also contribute to the residual 4-fermion LS LD interaction. In Table 1, we summarize basic physical quantities calculated in the NJL model together with the corresponding empirical values. The numerical values are obtained with the following parameter set; Λ = 631.4MeV, g Λ2 = 3.67, g Λ5 = 9.29, m = S D − s 135.7MeV, where we have used a three-momentum cutoff scheme. 2 The predicted constituent quark masses Mu = Md = 335MeV and Ms = 527MeV, (2.9) are consistent with the phenomenological masses extracted from the baryon magnetic moments [12].
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