Possible Color Octet Quark-Anti-Quark Condensate in the Instanton Model

Inspired by a recent proposal for a Higgs description of QCD we study the possible formation of a color-octet/flavor-octet quark-anti-quark condensate in the instanton liquid model. For this purpose we calculate two-point correlation functions of color-singlet and octet quark-anti-quark operators. We find long range order in the standard 〈ψ̄ψ〉 channel, but not in the color-octet channel. We emphasize that similar calculations in lattice QCD can check whether or not a color-flavor locked Higgs phase is realized in QCD at zero temperature and baryon density. It was recently argued that confinement in QCD can be understood in terms of an effective Higgs description where SU(3) gauge invariance is spontaneously broken by a coloroctet quark-anti-quark condensate [1–4] (see [5,6] for earlier speculations in this direction) 〈ψ̄(λ)C(λ A)TFψ〉 = 2 ( δ i δ b j − 1 3 δδij ) 〈ψ̄ i ψ b j〉. (1) Here, (λ)C (A=1, . . . , 8) is a color SU(3) matrix and (λ )F is a flavor SU(3) matrix. Color indices in the fundamental representation of SU(3) are denoted by a, b and flavor indices by 1 i, j. The condensate (1) breaks the local gauge symmetry as well as the SU(3)L × SU(3)R chiral symmetry but leaves a diagonal SU(3) unbroken. This symmetry acts as the physical flavor symmetry. The Higgs field (1) allows for a remarkably simple description of the gross features of the QCD spectrum. Gluons acquire a mass via the Higgs mechanism and carry the flavor quantum numbers and charges of the vector meson octet. The quark degrees of freedom transform as an octet and a heavier singlet and carry the quantum numbers of the baryon octet and singlet. This type of Higgs description was originally proposed as a complementary picture of QCD at large baryon chemical potential [7,8]. In that case, the primary Higgs field is given by the color-flavor locked diquark condensate 〈ψ i Cγ5ψ b j〉 = φ ( δ i δ b j − δ b i δ a j ) . (2) This condensate has the same residual color-flavor symmetry as (1) but also breaks the U(1) of baryon number. At large baryon density this is not a concern and simply corresponds to baryon superfluidity. The diquark condensate is dynamically generated by attractive interactions in the color anti-triplet quark-quark channel. At very large chemical potential this attraction is generated by one-gluon exchange [9] while at intermediate density instanton induced interactions are likely to play a role [10,11]. Because the color-octet quark-antiquark condensate (1) is consistent with the symmetries of the color-flavor locked diquark phase we expect this operator to have a non-zero expectation value in the high density phase [12–14]. This expectation is borne out by explicit calculations [12,13] but the value of the color-octet condensate is strongly suppressed with respect to the primary Higgs field, 〈ψ̄(λ)C(λ A)TFψ〉 ≃ 〈ψ̄ψ〉 ≪ 〈ψCγ5(λ )C(λ )Fψ〉. The suppression is due to an approximate Z2 symmetry in the high density phase. At zero baryon density the dynamical origin of a possible color-octet quark-anti-quark is unclear. Both one-gluon exchange and instantons are repulsive in this channel. Nevertheless, given the attractiveness of the scenario outlined in [1–4] it seems worthwhile to investigate this question beyond perturbation theory. In this note, we report on a calcu2 lation using the instanton liquid model [15,16]. This model correctly accounts for chiral symmetry breaking at zero baryon density and the formation of a diquark condensate at high density. The color-octet quark-anti-quark condensate at high density is also induced by instantons. Furthermore, Wetterich suggested that an instability in the instanton induced effective potential in three flavor QCD might be the dynamical origin of the color-octet quark-anti-quark condensate [3]. First we have to discuss how to identify the Higgs phase characterized by (1). The coloroctet condensate (1) is not a gauge invariant operator and therefore it cannot be used as an order parameter. The simplest alternative is the square of the octet condensate. This is similar to studying the vev of the square of the SU(2) Higgs field in the standard model. In the present case, however, this is not very useful. The square of the octet condensate receives large contributions from fluctuations associated with ordinary chiral symmetry breaking. This is suggested by the factorization approximation [17] which gives 〈 ( ψ̄(λ)C(λ A)TFψ )2 〉 ≃ 8 81 〈ψ̄ψ〉. (3) In the instanton model 〈(ψ̄(λ)C(λ A)TFψ) 〉 is about 2-3 times larger than this estimate, but even larger deviations from factorization are observed in other Lorentz-scalar four-quark operators. Instead of the square of the octet condensate we propose to study the coloroctet quark-anti-quark correlation function. If QCD can be described in terms of the Higgs picture advocated in [1] we expect to observe long range order in this correlation function. We should note that the color-octet correlation function is also not a gauge invariant object. This problem can be addressed by calculating the correlator in some fixed gauge or by inserting two gauge strings. One might worry that if the gauge strings are included the correlator will no longer approach a constant even if there is long range order. Nevertheless, if the Higgs description makes sense there has to be a clear difference between the color-octet correlator and a generic gauge invariant correlation function without long range order. We have calculated the correlation functions in the color-singlet/flavor-singlet, colorsinglet/flavor-octet and color-octet/flavor-octet channel. The correlation functions are de3 fined by Π[1][1](x, y) = −〈Tr [ S(x, y)S(y, x) ] 〉+ 〈Tr [S(x, x)] Tr [ S(y, y) ] 〉, (4) Π[1][8](x, y) = −〈Tr [