Chiral properties of the constituent quark model

We show that, in a model based exclusively on constituent-quark degrees of freedom interacting via a potential, the full axial current is conserved if the spectrum of Q̄Q states contains a massless pseudoscalar. The current conservation emerges nonperturbatively if the model satisfies certain constraints on (i) the axial coupling gA of the constituent quark and (ii) the Q̄Q potential at large distances. We define the chiral point of the constituent quark model as that set of values of the parameters (such as the masses of the constituent quarks and the couplings in the Q̄Q potential) for which the mass of the lowest pseudoscalar Q̄Q bound state vanishes. At the chiral point the main signatures of the spontaneously broken chiral symmetry are shown to be present, namely: the axial current is conserved, the decay constants of the excited pseudoscalar bound states vanish, and the pion decay constant has a nonzero value. Chiral symmetry is a basic symmetry of massless QCD which, apart from the axial anomaly in the flavor-singlet channel, entails the conservation of the axial-vector current. The masses of the light u and d quarks are small compared to the confinement scale, and consequently the chiral limit serves as a good approximation for the light-quark sector of QCD. Chiral symmetry in QCD is spontaneously broken, and is thus not a symmetry of the hadron spectrum: except for the existence of the octet of light pseudoscalar mesons, the lowest-energy part of the hadron spectrum shows no trace of chiral symmetry. Because of confinement, the calculation of the hadron mass spectrum directly from the QCD Lagrangian is a very challenging task, which requires a nonperturbative approach. QCD-inspired constituent quark models (i.e., models based on constituent-quark degrees of freedom in which mesons appear as Q̄Q bound states in a potential) proved to be quite successful for the description of the mass spectrum of hadrons and their interactions at low momentum transfers [1, 2, 3]. Because of the proper description of the hadron mass spectrum, the Lagrangian of the constituent quark model cannot be chirally invariant: it would produce a chirally invariant spectrum of hadron states. Consequently, the Noether axial current found in such models is not conserved but satisfies the divergence equation ∂ μ [Q̄(x)γμγ5Q(x)] = 2mQQ̄(x)iγ5Q(x). (1) In a recent paper [4] we have shown that, nevertheless, taking into account the infinite number of diagrams describing the Q̄Q interactions, leads to the full axial current of the constituent quarks, which turns out to have the structure 〈0|q̄γμ γ5q|Q̄Q〉 =gA(p) { Q̄γμγ5Q+2mQ pμ p2 Q̄γ5Q } +gA(p 2) pμ p2 Q̄γ5Q 2mQO(M π) p2 −M2 π . (2) Obviously, the term in curly brackets is transverse by virtue of Eq. (1). Therefore, the full “constituent-quark axial current” is conserved if the mass Mπ of the pion, the lowest Q̄Q pseudoscalar bound state, vanishes. As has been demonstrated in Ref. [4], to guarantee the axial current conservation up to terms of order O(M2 π) requires that the axial coupling gA of the constituent quarks is not constant but that it is related to the pion wave function Ψπ(s) by gA(s) = ηA(s−M π)Ψπ(s)+O(M π), ηA = const. (3) It should be recalled here that the spontaneous breaking of chiral symmetry requires not only the conservation of the axial current, but also the nonvanishing of the coupling of a massive fermion (such as a nucleon or a constituent quark) to the pion, i.e., gA(s) should be nonzero at s = M2 π . Consequently, to be compatible with the spontaneous breaking of chiral symmetry, the potential model should generate a light pseudoscalar bound state for which Ψπ(s = M2 π) has a pole at s = M2 π [4]. In Ref. [4] it is shown that the behavior of Ψπ(s) at s = M2 π is related to the behavior of the potential of the Q̄Q interaction at large separations r. More precisely, Ψπ(s) exhibits a pole at s = M2 π only if the potential saturates at large r: V (r → ∞) = const < ∞. (4) In this case the nearly massless pion is a strongly bound Q̄Q state with binding energy ε ≃ 2m. The observed conservation of the axial current allows us to define the chiral point of the constituent quark model as exactly that set of values of the parameters which leads to a massless lowest pseudoscalar Q̄Q bound state.1 In the following, let us consider certain properties of the constituent quark model at the chiral point. Decay constants of pseudoscalar mesons Making use of the relation (3) between the axial coupling of the constituent quark, gA(s), and the pion wave function, the standard quark-model expression for the decay constant of the n-th excitation of a pseudoscalar meson [3] takes the form [4] fP(n) = 2mQηA √ Nc ∫ dsΨ0(s)Ψn(s)ρ(s,mQ,m 2 Q) s−M2 π s +O(M π). (5) The wave functions Ψn(s) of the pseudoscalar states satisfy the orthogonality condition