Meson Mass at Large Baryon Chemical Potential in Dense QCD

We reexamine the quark mass induced term in chiral Lagrangian in color-flavor locking phase in dense QCD, and show that the meson mass term is determined by three independent invariants under chiral-axial symmetry, and is given in terms of the quark mass, gap, and the chemical potential by mπ ∼ mq∆/μ ln(μ2/∆2). Thus mesons become massless as μ → ∞. Cold, dense quark system with three massless quark flavors was proposed to be in color-flavor locking phase first by Alford, Rajagopal and Wilczek [1], and subsequently confirmed by others [2] , in which di-quark condensation occurs in a pattern < χaiχ b j >= − < φ̄ai φ̄bj >= k1 δ i δ j + k2 δ j δ i (1) where χi , φ a i , a the color index, i the flavor index, denote two-component Weyl fermions for the left-handed quarks and the complex conjugate of right-handed quarks, respectively. In this phase the symmetry of dense QCD, namely, the global color-chiral-axial-baryon number symmetry SUc(3)×SUL(3)×SUR(3)×UA(1)×UB(1) is spontaneously broken to Email: [email protected] Email: [email protected] Email: [email protected] 1 SUc+L+R(3) by the quark condensation. The UA(1), which is anomalous at zero density, is a good symmetry at high density, since instanton effects are screened out at large baryon chemical potential μ [3]. Under the symmetry quarks transform as χai → ULijgeχj φai → U Rijgeφj (2) where UL,R ∈ SUL,R(3), g ∈ SUc(3) and e, e for the axial and baryon number symmetry, respectively. Upon the quark condensation 9 + 9 Nambu-Goldstone bosons are generated, each 9 coming from the χχ and φ̄φ̄ condensation. In nonlinear realization of the symmetry these Nambu-Goldstone bosons can be represented by the unitary matrices U i , V a i which parameterize the coset space SUc(3)×SUL(3)×UA+B(1)/SUc+L(3) and SUc(3)×SUR(3)× UB−A(1)/SUc+R(3), respectively, and transform as U i → ULijgU b j e V a i → URijgV b j e. (3) Upon gauging the color symmetry gluons absorb 8 Nambu-Goldstone bosons, becoming massive mAμ ∼ gFπ, where g is the gauge coupling, via Higgs mechanism, and there remain 10 Nambu-Goldstone bosons (mesons) in the physical spectrum. Now when quarks receive small mass via the Dirac mass term Lmq = −mijχaiφaj + h.c. (4) these mesons, except for the one associated with the baryon number symmetry, become massive due to the explicit breaking of the chiral-axial symmetry. Since these mesons would be the lowest lying excitations in cold, dense system such as neutron star their spectrum is of considerable interest and was investigated recently by several groups [6, 7, 8]. A notable feature obtained from the investigation is that at large chemical potential meson mass is independent of the gap, mπ ∼ mq , where mq denotes quark mass, and when there is no sextet condensation (i.e. k1 = −k2) meson mass is determined by a