Chiral symmetry breaking with lattice propagators

We study chiral symmetry breaking using the standard gap equation, supplemented with the infrared-finite gluon propagator and ghost dressing function obtained from large-volume lattice simulations. One of the most important ingredients of this analysis is the non-Abelian quark-gluon vertex, which controls the way the ghost sector enters into the gap equation. Specifically, this vertex introduces a numerically crucial dependence on the ghost dressing function and the quark-ghost scattering amplitude. This latter quantity satisfies its own, previously unexplored, dynamical equation, which may be decomposed into individual integral equations for its various form factors. In particular, the scalar form factor is obtained from an approximate version of the “one-loop dressed” integral equation, and its numerical impact turns out to be rather considerable. The detailed numerical analysis of the resulting gap equation reveals that the constituent quark mass obtained is about 300 MeV, while fermions in the adjoint representation acquire a mass in the range of (750–962) MeV.

Figure 1

The full fermion-gluon vertex.

Figure 2

Diagrammatic representation of the fermion-ghost scattering kernel $H(p_1,p_2,p_3)$.

Figure 3

The SDE for the fermion propagator given by Eq. (2.11). The gray blobs represent the fully-dressed gluon and quark propagators, while the black blob denotes the dressed fermion-gluon vertex.

Figure 4

Diagrammatic representation of the quark-ghost scattering kernel, $H(-q/2,-q/2,q)$, at one-loop.

Figure 5

Lattice results for the gluon propagator, $\Delta (q)$, and ghost-dressing, $F(q)$, obtained in Ref. 18 and renormalized at $\mu =4.3\mathrm{GeV}$. Left panel: The continuous line represents the gluon lattice data fitted by Eq. (4.1) when $m=520\mathrm{MeV}$, $g_1^2=5.68$, ${\rho }_1=8.55$, and ${\rho }_2=1.91$. Right panel: The lattice data for $F(q)$ fitted by Eq. (4.3) using $g_2^2=8.57$, $m=520\mathrm{MeV}$, ${\rho }_3=0.25$, and ${\rho }_4=0.68$.

Figure 6

Numerical result for the form factor $X_0^{[1]}(q)$ of the quark-ghost scattering kernel given by Eq. (3.17) when $\alpha ({\mu }^2)=0.295$.

Figure 7

Left panel: The individual contribution of the ingredients composing ${\mathcal{K}}_0(q)$. The green dotted-dashed line represents the case where ${\mathcal{K}}_0(q)=\Delta (q)$, the blue dotted line the case where ${\mathcal{K}}_0(q)=\Delta (q)F(q)$, in the red dashed curve ${\mathcal{K}}_0(q)=\Delta (q)F^2(q)$ and, finally the black continuous line represents the case where ${\mathcal{K}}_0(q)$ assumes the full form used in our calculation ${\mathcal{K}}_0(q)=\Delta (q)F^2(q)X_0^{[1]}(q)$. Right panel: The corresponding dynamical quark mass generated when we use in Eqs. (3.23, 3.24) the different forms of ${\mathcal{K}}_0(q)$ are shown in the left panel.

Figure 8

Left panel: The numerical solution for the quark wave-function $A^{-1}(p^2)$ in the fundamental representation when the non-Abelian versions of BC (red dashed curve) and CP (black continuous curve) vertices. are employed Right panel: The numerical solution for the dynamical quark mass $\mathcal{M}(p^2)$. The red dashed curve represents the dynamical mass generated when the BC vertex is used, while the black continuous line is the solution found with the CP vertex.

Figure 9

Left panel: The numerical solution for the fermion wave-function $A_{\mathrm{adj}}^{-1}(p^2)$ in the adjoint representation when the modified BC vertex (dotted red curve) and the modified CP vertex (black continuous curve) are employed. Right panel: The numerical solution for the dynamical fermion mass ${\mathcal{M}}_{\mathrm{adj}}(p^2)$. The dotted red curve represents the dynamical mass generated when the BC vertex is used. The black continuous line is the solution obtained with the CP vertex.