Quantum Mechanics of Confinement and Chiral Symmetry Breaking in Two-dimensional Qcd

The system of light quark and heavy anti-quark source is studied in 1+1 QCD in the large NC limit. Making use of the modified Fock–Schwinger gauge allows to consider simultaneously the spectroscopical problem of the qQ̄ bound states and the problem of the light quark Green function. The Dirac-type equation for the spectrum of the system is proved to be equivalent to the well-known ’t Hooft one in the one body limit. The unitary transformation from the Dirac–Pauli representation to the Foldy–Wouthuysen one is carried out explicitly, and it is shown that the equation in the Foldy–Wouthuysen representation can be treated as a gap equation which defines the light quark self-energy in the modified Fock– Schwinger gauge. The Foldy–Wouthuysen angle is found to play the role of the Bogoliubov–Valatin one and to give the standard value of the chiral condensate. Connections of the given formalism to the standard four-dimensional QCD are outlined and discussed. For the first time the two-dimensional model of QCD in the limit of infinite number of colours NC was considered in 1974 by ’t Hooft 1 and the celebrated equation of the same name was derived in the light-cone gauge. Four years later, in 1978, this equation was re-derived in the axial gauge. So the model seems to have been studied well enough. Still it attracts considerable attention as a problem with features very much similar to those of standard four-dimensional QCD. Besides the usual assumption NC →∞ limit that allows to sum up only planar diagrammes, we make use of the so-called modified Fock-Schwinger or Balitsky gauge Aa1(x0, x) = 0 A a 0(x0, 0) = 0 (1) As soon as gauge (1) is a kind of radial one, the gluonic field can be expressed in terms of the field strength tensor that yields the gluon propagator in the form K 00(x0 − y0, x, y) = δ ab g 2 2 δ(x0 − y0)(|x− y| − |x| − |y|) ≡ δ K(x, y), (2) and other components equal to zero. Note that K can be naturally broken into local (K ∼ |x − y|δ(x0 − y0)) and non-local (K ∼ (|x|+ |y|)δ(x0 − y0)) parts. Green function for the q − Q̄ system under consideration has the form ∗ SqQ̄(x, y) = 1 NC ∫ DψDψ̄DAμ exp { − 1 4 ∫ dxF a2 μν − ∫ dxψ̄(i∂̂ −m− Â)ψ } × ×ψ̄(x)SQ̄(x, y|A)ψ(y), (3) where anti-quark Green function SQ̄ is introduced. The main advantage of our peculiar gauge choice is the fact that the anti-quark is decoupled completely so that SQ̄ can be substituted in a very simple form: SQ̄(x, y|A) = SQ̄(x− y); SQ̄ = −i ( 1 + γ0 2 θ(t)e + 1− γ0 2 θ(−t)e ) δ(x). (4) On integrating gluon degrees of freedom in (3), we arrive at the effective Lagrangian for the light quark which leads in turn to the Schwinger–Dyson equation (i∂̂x −m)S(x, y) + iNC 2 ∫ dzγ0S(x, z)γ0K(x, z)S(z, y) = δ (x− y), (5) where S(x, y) = 1 NC S α(x, y). (6) It is very instructive to note here that the role played by Green function (6) is twofold. By construction S is the Green function of the light quark, but due to a very passive part of the static anti-quark it plays the role of the Green function of the whole qQ̄ system as well, so that the problem of the light quark Green function and the spectroscopical problem for the qQ̄ system can be considered simultaneously. We shall get back to this statement later on while discussing the chiral properties of the model. Approach based on spectral decomposition of the Green function (6) turns out very useful †, so one has S(q10, q1, q20, q2) = 2πδ(q10−q20) ∑ εn>0 φ n (q1)φ̄ (+) n (q2) q10 − εn + ∑ εn<0 φ n (q1)φ̄ (−) n (q2) q10 + εn  . (7) To proceed further we assume that the Foldy-Wouthuysen operator T (p) = e 1 2 θF (p)γ1 diagonalizing equation (5) exists and that angle θF is the same for all n. With such an assumption applied Schwinger–Dyson equation (5) reduces to the Dirac-type equation in the Hamiltonian form (f = g NC 4π , α = γ0γ1, β = γ0, ): (αp+βm)φn(p)−πf ∫ dqdk(βcosθF (q)+αsinθF (q))K(p−q, k−q)φ 0 n(k) = Enφ 0 n, (8) ∗We adopt the following γ-matrix convention: γ0 = σ3, γ1 = iσ2, γ5 = σ1 †The approach based on diagrammatic technics leads to the same results where φn(p) being scalar wave function φ n (p) = φ 0 n(p)T (p) ( 1 0 ) , φ n (p) = φ 0 n(p)T (p) ( 0 1 ) . (9) Let us consider only local (i.e. generated by K) part of interaction which reduces to a mass operator Σ and can be naturally parametrized via two scalar functions E(p) and θ(p) in the convenient form Σ(p) ≡ [E(p)cosθ(p)−m] + γ1 [E(p)sinθ(p)− p] . (10) Self-consistency condition for such a parametrization makes E(p) and θ(p) satisfy a system of coupled equations:  E(p)cosθ(p) = m+ 2 ∫ – dk (p− k) cosθ(k) E(p)sinθ(p) = p+ 2 ∫ – dk (p− k) sinθ(k), (11) It is easy to verify that if we identify the Foldy-Wouthuysen angle θF with the Bogoliubov–Valatin one θ‡ then the non-local interaction diagonalizes as well, so that the Foldy-Wouthuysen representation of equation (8) takes the Schrödinger-type form εnφ 0 n(p) = E(p)φ 0 n(p)− f ∫ – dk (p− k)2 cos θ(p)− θ(k) 2 φn(k). (12) Equation (12) is nothing but the one-body limit of the well-known ’t Hooft equation. As mentioned above, Green function of the qQ̄ system constructed from the solutions of equation (12) is the one of the light quark as well, so that the chiral condensate can be easily calculated < q̄q >= −i T r x→y+ S(x, y) = − NC π ∫ ∞