Dependence of the Landau gauge ghost-gluon-vertex on the number of flavors

The gauge-boson, ghost and fermion propagators as well as the gauge-boson-ghost vertex function are studied for $\mathrm{SU}(N)$, $\mathrm{Sp}(2N)$, and $\mathrm{SO}(N)$ gauge groups. We solve a set of coupled Dyson-Schwinger equations in Landau gauge for a variable, fractional number $N_f$ of massless fermions in the fundamental representation. For large $N_f$ we find a phase transition from a chirally broken into a chirally symmetric phase that is consistent with the behavior expected inside the conformal window. Even in the presence of fermions the gauge-boson-ghost-vertex dressing remains small. In the conformal window this vertex shows the expected power law behavior. It does not assume its tree-level value in the far infrared, but the respective dressing function is a constant greater than 1.

Figure 1

Truncated DSEs for the propagators (quark, ghost, and gluon from top to bottom) and the ghost-gluon vertex. Thick blobs denote dressed vertices whereas small blobs denote bare vertices. All internal lines are fully dressed. The $N_c$-dependent prefactors for quarks in the fundamental representation are given in Table 1.

Figure 2

Dressing functions of the propagators for different $N_f$ with and without a dynamical ghost-gluon vertex in QCD obtained from the truncated set of DSEs. The quark mass function’s value at vanishing momenta, $M_f(p^2\to 0)$, is an order parameter for chiral symmetry breaking. For $N_f=5$ chiral symmetry is restored, and the Yang-Mills propagators obey power laws in the far IR.

Figure 3

Ghost-gluon vertex for $N_f=0$, 2, 5 in QCD obtained from the truncated set of DSEs. The larger the number of quarks the closer the vertex’ dressing function gets to its tree-level value of 1. For $N_f=5$ the soft-limit value of the vertex does not assume its tree-level value one.

Figure 4

Infrared value of $M_f(p^2\to 0)$ as a function of $N_f$. Below the onset of the conformal window we always show the decoupling type solutions which leads to a smaller value of $N_f$ for which no fermion masses are dynamically generated (see the discussion in Sec. 3). At low values of $N_f$ the system becomes numerically unstable. This can be attributed to cancellations of the loop diagrams in the gluon DSE—see Sec. 4b.

Figure 5

The gluon loop ${\tilde{\mathrm{\Pi }}}_{\text{gluon}}(p^2)\equiv -{\mathrm{\Pi }}_{\text{gluon}}(p^2)/{\mathrm{\Pi }}_{\text{ghost}}(p^2)$ and quark loop ${\tilde{\mathrm{\Pi }}}_{\text{quark}}(p^2)\equiv -{\mathrm{\Pi }}_{\text{quark}}(p^2)/{\mathrm{\Pi }}_{\text{ghost}}(p^2)$ of the gluon DSE, relative to the ghost loop, ${\mathrm{\Pi }}_{\text{ghost}}(p^2)$ in $SU(3)$ with fundamental fermions. The dashed lines at one indicates the ghost loop. The sum of quark and gluon loop has large cancellations with the ghost loop. Even for small $N_f$ the quark loop becomes quickly significant. At $N_f\approx 0.8$ the gluon loop changes its sign in the infrared and midmomentum regime.

Figure 6

Ghost-gluon vertex for orthogonal momenta and for the gauge groups Sp(4) (upper panel) and SO(8) (lower panel) for different values of $N_f$.

Figure 7

Ghost-gluon vertex for $N_f=2$ for the gauge groups SU(3), Sp(4), and SO(8) for orthogonal momenta.

Figure 8

Ghost-gluon vertex for orthogonal momenta and for the gauge groups $\mathrm{SU}(N)$, $N=2$, 3, 4, 5 and $N_f=2$ (upper panel) and $\mathrm{SU}(N)$, $N=2$, 3, 4, 5 and $N_f=N$.