Spontaneous breaking of chiral symmetry for confining potentials

Abstract

Using the Bogoliubov-Valatin variational method, we show that the chiral-invariant vacuum is unstable for a color, fourth-component vector powerlike potential r a ( 0 <a < 3 ) independently of the strength of the coupling constant. The fermion self-energy is negative and dominates over the positive potential energy, destabilizing the vacuum by $$ pair condensation. This self-energy is finite but infrared singular, reflecting the behavior of the potential at large distances. We give an analytical proof of the fact that the energy of the unbroken vacuum is not minimum. The proof extends to logarithmic potentials as a+O, but breaks down for a 2 3 (number of spatial dimensions) due to severe infrared singularities. If the confining potential possesses a spin-spin piece, there are critical values of its strength, depending on the power a, beyond which the stability of the chiral-invariant vacuum is restored. In the case of the harmonic oscillator a=2, the gap equation reduces to a nonlinear second-order differential equation. We find (besides the usual chiral degeneracy) an infinite number of solutions breaking chiral symmetry, higher in energy as the number of their nodes increases. We compute the expectation value of $$ and the mass gap for the new vacuum, the lowest solution in energy. The infrared singularity of the massless fermion self-energy is removed for the stable broken solution.

Bounds on the derivatives of the Isgur–Wise function from sum rules in the heavy quark limit of QCD

“…Therefore, the functions τ (1) 1/2 (w) and τ (1) 3/2 (w) correspond, respectively, to the functions ζ(w) and τ (w) defined by Leibovich et al [4]. But in all this paper we will keep the general notation τ (ℓ)(n) ℓ+1/2 (w), τ (ℓ)(n) ℓ−1/2 (w).…”

“…[1] the IW functions τ (ℓ)(n) ℓ+1/2 (w) and τ (ℓ)(n) ℓ−1/2 (w), that correspond to the orbital angular momentum ℓ of the light quark relative to the heavy quark, j = ℓ ± 1 2 being the total angular momentum of the light cloud. We could made explicit the contribution of the states 1…”

Analytical approach to chiral symmetry breaking in Minkowski space

“…In the literature the mass gap equation is usually solved either in equal time [3] or in Euclidean space [4]. In the equal time approximation, the Lorentz invariance is lost.…”

“…SχSB also implies the Partially Conserved Axial Current theorems [2], including a small mass for the pion. In the literature the mass gap equation is usually solved either in equal time [3] or in Euclidean space [4], in order to avoid the poles and complex quantities which are expected in full Minkowsky calculations. Recently the scientific community has been exploring different approaches to the Minkowsky space [5].…”

The effect of nonlocal confining kernels on magnetic chiral condensates

“…There are several approaches one can adopt to obtain the mass gap equation. It can be derived as the condition for the vacuum energy to be a minimum [8,9], or in the form of a Dyson equation for the fermion propagator [9], or as a Ward identity [10]. Here we shall derived it by imposing the vanishing of the anomalous (nondiagonal) terms in the Hamiltonian, corresponding to the direct creation (or annihilation) of a fermion-antifermion pair [11,12].…”