Dynamical chiral symmetry breaking and a critical mass

On a bounded, measurable domain of non-negative current-quark mass, realistic models of the QCD gap equation can simultaneously admit two nonequivalent dynamical chiral symmetry breaking (DCSB) solutions and a solution that is unambiguously connected with the realization of chiral symmetry in the Wigner mode. The Wigner solution and one of the DCSB solutions are destabilized by a current-quark mass, and both disappear when that mass exceeds a critical value. This critical value also bounds the domain on which the surviving DCSB solution possesses a chiral expansion. This value can therefore be viewed as an upper bound on the domain within which a perturbative expansion in the current-quark mass around the chiral limit is uniformly valid for physical quantities. For a pseudoscalar meson constituted of equal-mass current quarks, it corresponds to a mass $m_{0^{-}}~0.45$ GeV. In our discussion, we employ properties of the two DCSB solutions of the gap equation that enable a valid definition of $\langle \overline{q}q\rangle$ in the presence of a nonzero current mass. The behavior of this condensate indicates that the essentially dynamical component of chiral symmetry breaking decreases with increasing current-quark mass.

Figure 1

Zeros of $G(M)$ give the solution of the gap equation defined by Eqs. (13) and (15). Solid curve, obtained with $m^{\mathrm{bm}}=0$, in which case $G(M)$ is odd under $M\to -M$; long-dashed curve, $m^{\mathrm{bm}}=0.01$; short-dashed curve, $m^{\mathrm{bm}}=m_{\mathrm{cr}}^{\mathrm{bm}}=0.033$; dotted curve, $m^{\mathrm{bm}}=0.05$. All dimensioned quantities in units of $\tilde{\Lambda }$.

Figure 2

Circles: Reciprocal of the $n$th roots of the coefficients in a chiral expansion of $M_{+}$ around $m^{\mathrm{bm}}=0$, Eq. (20), as a function of $t=1/n$. Dashed curve: the function in Eq. (23). Dotted curve: $m_{\mathrm{cr}}^{\mathrm{bm}}=0.033$. All dimensioned quantities in units of $\tilde{\Lambda }$.

Figure 3

Evolution with current-quark mass of $\overline{M}$ (solid curve) and $\check{M}$ (dashed curve). All dimensioned quantities in units of $\tilde{\Lambda }$.

Figure 4

Evolution with current-quark mass $m({\zeta }_{19})$ of $A(p^2=0,{\zeta }^2)$ (dimensionless), $B(p^2=0,{\zeta }^2)$ (GeV) as calculated with $\omega =0.4$ GeV in Eq. (29): solid curve, $B_{+}(0)$; dash-dot curve, $A_{+}(0)$; long dashed curve, $B_{-}(0)$; sparse dotted curve, $A_{-}(0)$; short dashed curve, $B_W(0)$; dense dotted curve, $A_W(0)$.

Figure 5

Momentum dependence of the dressed-quark mass function $M(p^2)$: upper panel, chiral limit; middle panel, $m({\zeta }_{19})=5$ MeV; lower panel, $m({\zeta }_{19})=50$ MeV, for which there is naturally no $M_{-}(p^2)$ solution. In each panel, the solid curve is $M_{+}(p^2)$; dashed curve, $M_{-}(p^2)$; dotted curve, $M_W(p^2)$. All results obtained with $\omega =0.4$ GeV.

Figure 6

Momentum dependence of the dressed-quark mass-function: dashed curve, $M_{-}(p^2)$; dotted curve, $M_W(p^2)$; both obtained with $m({\zeta }_{19})=5$ MeV and $\omega =0.4$ GeV.

Figure 7

Evolution with current-quark mass $m({\zeta }_{19})$ of the massive-quark condensate defined in Eq. (37), calculated with $\omega =0.4$ GeV and at the renormalization point $\zeta =19$ GeV.