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Light and Strange Baryons, Two-Baryon Systems and the Chiral Symmetry of QCD L

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Light and Strange Baryons, Two-Baryon Systems and the Chiral Symmetry of QCD L 1 Light and strange baryons, two-baryon systems and the chiral symmetry of QCD L. Ya. Glozmanab aInstitute for Theoretical Physics, University of Graz, A-8010 Graz, Austria bDepartment of Physics, University of Helsinki, POB9, FIN-00014 Helsinki, Finland Beyond the scale of spontaneous breaking of chiral symmetry light and strange baryons should be considered as systems of three constituent quarks with confining interaction and a chiral interaction that is mediated by Goldstone bosons between the constituent quarks. The flavor-spin structure and sign of the short-range part of the Goldstone boson exchange interaction reduces the SU(6)FS symmetry down to SU(3)F × SU(2)S, induces hyperfine splittings and provides correct ordering of the lowest states with positive and negative parity. A unified description of light and strange baryon spectra calculated in a semirelativistic framework is presented. It is demonstrated that the same short-range part of the Goldstone boson exchange between the constituent quarks induces a strong short- range repulsion in NN system when the latter is treated as 6Q system. Similar to the NN system there should be a short-range repulsion in other NY and YY two-baryon systems. We also find that the compact 6Q system with the ”H-particle” quantum numbers lies a few hundreds MeV above the ΛΛ threshold. It then suggests that the deeply bound H-particle should not exist. 1. Spontaneous Breaking of Chiral Symmetry and its Implications At low temperatures and densities the SU(3)L × SU(3)R chiral symmetry of QCD Lagrangian is spontaneously broken down to SU(3)V by the QCD vacuum (in the large Nc limit it would be U(3)L × U(3)R → U(3)V). A direct evidence for the spontaneously broken chiral symmetry is a nonzero value of the quark condensates for the light flavors < |qq¯ | >≈−(240 − 250MeV)3, which represent the order parameter. That this is indeed so, we know from three sources: current algebra, QCD sum rules, and lattice gauge calculations. There are two important generic consequences of the spontaneous breaking of chiral symmetry (SBCS). The first one is an appearance of the octet of pseudoscalar mesons of low mass, π, K,η, which represent the associated approximate Goldstone bosons 0 (in the large Nc limit the flavor singlet state η should be added). The second one is that valence (practically massless) quarks acquire a dynamical mass, which have been called historically constituent quarks. Indeed, the nonzero value of the quark condensate itself implies at the formal level that there should be rather big dynamical mass, which could be in general a moment-dependent quantity [1]. Thus the constituent quarks should be considered as quasiparticles whose dynamical mass comes from the nonperturbative gluon 2 and quark-antiquark dressing. The flavor-octet axial current conservation in the chiral limit tells that the constituent quarks and Goldstone bosons should be coupled with the strength g = gAM/fπ [2], which is a quark analog of the famous Goldberger-Treiman relation. We cannot say at the moment for sure what is the microscopical mechanism for SBCS in QCD. Any sufficiently strong scalar interaction between quarks will induce the SBCS (e.g. the instanton - induced interaction contains the scalar part, or it can be generated by monopole condensation, etc.). The most general aspects of SBCS such as constituent quark mass generation, Goldstone boson creation and their couplings are well illustrated by the Nambu and Jona-Lasinio model [3]. For the low-energy baryon properties it is only essential that beyond the spontaneous chiral symmetry breaking scale (i.e. at low resolution) new dynamical degrees of freedom appear - constituent quarks and chiral fields which couple together. The low-energy baryon properties are mainly determined by these dynamical degrees of freedom and the confining interaction. We have recently suggested [4] that in the low-energy regime light and strange baryons should be considered as a system of 3 constituent quarks with an effective Q−Q interaction that is formed of central confining part and a chiral interaction mediated by the Goldstone bosons between constituent quarks. This physical picture allows to understand a structure of the baryon spectrum and to solve, in particular, the long-standing problem of ordering of the lowest baryons with positive- and negative-parity. 2. The Goldstone Boson Exchange Interaction and the Structure of Light and Strange Baryon Spectra The coupling of the constituent quarks and the pseudoscalar Goldstone bosons will ¯ ~ F µ~ (in the SU(3)F symmetric approximation) have the form g/(2m)ψγµγ5λ · ψ∂ φ within ¯ ~ F ~ the nonlinear realization of chiral symmetry (it would be igψγ5λ · φψ within the linear σ-model chiral symmetry representation). A coupling of this form, in a nonrelativistic reduction for the constituent quark spinors, will – to lowest order – give rise the ∼ ~σ · ~q~λF structure of the meson-quark vertex, where ~q is meson momentum. Thus, the structure of the potential between quarks ”i”and”j” in momentum representation is ~F ~F 2 2 2 V (~q) ∼ ~σi · ~qσj · ~q λi · λj D(q )F (q ), (1) where D(q2) is dressed Green function for chiral field which includes both nonlinear terms of chiral Lagrangian and fermion loops, F (q2) is meson-quark formfactor which takes into account the internal structure of quasiparticles. At big distances (~q → 0), one has 2 2 2 2 −1 2 D(q ) → D0(~q )=−(~q +µ) 6=∞and F (q ) → 1. It then follows from (1) that V (~q = 0) = 0, which means that the volume integral of the Goldstone boson exchange (GBE) interaction should vanish, d~rV (~r)=0. (2) Z This sum rule is not valid in the chiral limit, however, where µ = 0 and, hence, D0(~q = 0) = ∞. Since at big interquark separations the spin-spin component of the pseudoscalar- e−µr exchange interaction is V (r) ∼ r , it then follows from the sum rule above that at short 3 interquark separations the spin-spin interaction should be opposite in sign as compared to the Yukawa tail and very strong. It is this short-range part of the Goldstone boson exchange (GBE) interaction between the constituent quarks that is of crucial importance for baryons: it has a sign appropriate to reproduce the level splittings and dominates over the Yukawa tail towards short distances. In a oversimplified consideration with a free Klein-Gordon Green function instead of the dressed one in (1) and with F (q2) = 1, one obtains the following spin-spin component of Q − Q interaction: 2 −µr g 1 1 ~F ~F 2e V (r)= ~σi ·~σjλi ·λj{µ − 4πδ(~r)}. (3) 4π34mimj r In the chiral limit only the negative short-range part of the GBE interaction survives. Consider first, for the purposes of illustration, a schematic model which neglects the ra- dial dependence of the potential function V (r) in (3), and assume a harmonic confinement among quarks as well as mu = md = ms.Inthismodel ~ F ~ F Hχ=− Cχ λi · λj ~σi · ~σj. (4) i<j X Note, that contrary to the color-magnetic interaction from perturbative one-gluon ex- change, the GBE interaction is explicitly flavor-dependent. It is this circumstance which allows to solve the long-standing problem of ordering of the lowest positive-negative parity states. If the only interaction between the quarks were the flavor- and spin-independent har- monic confining interaction, the baryon spectrum would be organized in multiplets of the symmetry group SU(6)FS ×U(6)conf . In this case the baryon masses would be determined solely by the orbital structure, and the spectrum would be organized in an alternative sequence of positive and negative parity states, i.e. in this case the spectrum would be: ground state of positive parity (N = 0 shell, N is the number of harmonic oscillator exci- tations in a 3-quark state), first excited band of negative parity (N = 1), second excited band of positive parity (N = 2), etc. The Hamiltonian (4), within a first order perturbation theory, reduces the SU(6)FS × U(6)conf symmetry down to SU(3)F × SU(2)S × U(6)conf , which automatically implies a splitting between the octet and decuplet baryons (e.g. the ∆ resonance becomes heavier than nucleon). Let us now see how the pure confinement spectrum above becomes modified when the GBE Hamiltonian (4) is switched on. For the octet states N, Λ, Σ, Ξ (N = 0 shell) as well as for their first radial excitations of positive parity N(1440), Λ(1600), Σ(1660), Ξ(?) (N = 2 shell) the expectation value of the Hamiltonian (4) is −14Cχ. For the decuplet states ∆, Σ(1385), Ξ(1530), Ω (N = 0 shell) the corresponding matrix element is −4Cχ. In the negative parity excitations (N = 1 shell) in the N, Λ and Σ spectra (N(1535) - N(1520), Λ(1670) - Λ(1690) and Σ(1750) - Σ(?)) the contribution of the interaction (4) is −2Cχ. The first negative parity excitation in the Λ spectrum (N = 1 shell) Λ(1405) - Λ(1520) is flavor singlet and, in this case, the corresponding matrix element is −8Cχ. The latter state is unique and is absent in other spectra due to its flavor-singlet nature. These matrix elements alone suffice to prove that the ordering of the lowest positive and negative parity states in the baryon spectrum will be correctly predicted by the 4 chiral boson exchange interaction (4). The constant Cχ may be determined from the N−∆ splitting to be 29.3 MeV. The oscillator parameterhω ¯ , which characterizes the effective confining interaction, may be determined as one half of the mass differences 1 + between the first excited 2 states and the ground states of the baryons, which have the same flavor-spin, flavor and spin symmetries (e.g.
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