Chiral Symmetry Restoration and the Georgi Vector Limit

The chiral phase transition at high temperature and/or density is interpreted in terms of Georgi’s vector limit. We discuss three cases as possible support for this scenario: Quark-number susceptibility, cool kaons in heavy-ion process and an instanton-molecule picture for chiral restoration. Both the notion of “mended symmetry” and the Georgi vector limit are suggested to be relevant in nuclear physics of dense matter. One of the most intriguing – and largely unsettled – problems in strong interaction physics is how chiral symmetry is restored in matter as it becomes hot and/or dense. In this letter, we propose that the symmetry restoration involves the “vector limit” put forward by Georgi some years ago[1]. Georgi envisaged the vector limit to be appropriate in the large Nc limit where Nc is the number of colors and supposed that Nature with Nc = 3 is close to the limit, so that the departure from the limit could be treated as a perturbation. Here we are proposing that the vector limit is appropriate in QCD at the chiral phase transition in hot and/or dense hadronic matter. As support to our proposal, we shall discuss three cases: 1) lattice gauge calculations on quark-number susceptibility at high temperature; 2) “cool” kaons observed in heavy-ion collisions; 3) an instanton-molecule model for chiral phase transition. The Georgi vector limit We begin by sketching the essential points of the Georgi vector symmetry and vector limit[1]. Consider two chiral flavors u(p) and d(own), with chiral symmetry SU(2)L × SU(2)R. The standard way of looking at this symmetry is that it is realized either in NambuGoldstone (or Goldstone in short) mode, with SU(2)L×SU(2)R broken down spontaneously to SU(2)L+R or in Wigner-Weyl (or Wigner in short) mode with parity doubling. Georgi observes, however, that there is yet another way of realizing the symmetry which requires both Goldstone mode andWigner mode to co-exist. Now the signature for any manifestation of the chiral symmetry is the pion decay constant fπ 〈0|Aμ|π(q)〉 = ifπqμδ (1) whereAμ is the isovector axial current. The Goldstone mode is characterized by the presence of the triplet of Goldstone bosons, πi with i = 1, 2, 3 with a non-zero pion decay constant. The Wigner mode is realized when the pion decay constant vanishes, associated with the absence of zero-mass bosons. In the latter case, the symmetry is realized in a parity-doubled mode. The Georgi vector symmetry we are interested in corresponds to the mode (1) coexisting with a triplet of scalars Si with fS = fπ where 〈0|V i μ|S(q)〉 = ifSqμδ (2) where V i μ is the isovector-vector current. In this case, the SU(2) × SU(2) symmetry is unbroken. At low T and/or low density, low-lying isovector-scalars are not visible and hence either the vector symmetry is broken in Nature with fS 6= fπ or they are “hidden” in the sense that they are eaten up by vector particles (à la Higgs). In what follows, we would