Chiral symmetry breaking in continuum QCD

We present a quantitative analysis of chiral symmetry breaking in two-flavor continuum QCD in the quenched limit. The theory is set up at perturbative momenta, where asymptotic freedom leads to precise results. The evolution of QCD towards the hadronic phase is achieved by means of dynamical hadronization in the nonperturbative functional renormalization group approach. We use a vertex expansion scheme based on gauge-invariant operators and discuss its convergence properties and the remaining systematic errors. In particular, we present results for the quark propagator, the full tensor structure and momentum dependence of the quark-gluon vertex, and the four-Fermi scatterings.

Figure 1

FRG results in comparison to lattice QCD. Dimensionful quantities in Bowman et al., [20, 21], are rescaled by a factor 0.91 in both plots to match the scales of Sternbeck et al., [22] and the FRG results, [15, 19]. (a) Yang-Mills FRG gluon dressing function, (B1), taken from [15, 19] in comparison to quenched lattice data [20, 22]. (b) Our result for quark propagator dressing functions, (B6), in comparison to quenched lattice results [21] and mass function in GeV.

Figure 2

Dynamical hadronization: four-Fermi coupling (${\lambda }_{\pi }$, see Appendix pp2-s2c) vs corresponding coupling from dynamical hadronization, $(k{h}_{\pi }{)}^2/(2m_{\pi }^2)$, in a momentum-independent approximation similar to [3].

Figure 3

Pictorial description of our truncation for the effective action; see Fig. 7 for corresponding RG flows.

Figure 4

Strength of tensor structures and vertices. (a) Coupling strength, (17) and (B9), of quark-gluon vertex tensor components at symmetric point. Black: classical tensor structure, grey: chirally symmetric non-classical tensor structures, red: tensor structures that break chiral symmetry. (b) Running couplings ${\alpha }_s$ at symmetric point $p_1^2=p_2^2={(p_1+p_2)}^2$ from different vertices which takes the gluon gap into account analogous to (17) for the quark-gluon vertex. Critical gauge coupling for chiral symmetry breaking ${\alpha }_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}$.

Figure 5

RG-scale dependence of four-Fermi interactions and Yukawa coupling. (a) Renormalisation group scale dependence of dimensionless four- fermi interactions, see Appendix pp2-s2c, and bosonised $\sigma$-$\pi$ channel. Grey: respects chiral symmetry, blue: breaks ${U(1)}_A$, red: breaks $S{U(2)}_A$, magenta: breaks ${U(2)}_A$. (b) Independence of infrared model parameters from initial value and scale demonstrated for Yukawa coupling in a momentum-independent approximation similar to [3].

Figure 6

Normalized difference $(z_{\overline{q}Aq}^{(7)}-2z_{\overline{q}Aq}^{(5)})/z_{\overline{q}Aq}^{(7)}$ (blue solid) and $(z_{\overline{c}Ac}-z_{\overline{q}Aq}^{(1)})/z_{\overline{c}Ac}$ (red dashed). Small values indicate that the mSTI is applicable for constraining transversal tensor structures.

Figure 7

Types of diagrams that contribute to flow of propagators and vertices. Quark (solid lines), gluon (wiggly lines), ghost (dotted lines) and meson (dashed lines) propagators as well as the (1PI) vertices are dressed. Each diagram represents a sum of diagrams with (anti)symmetric permutations and regulator function inserted once in each internal propagator line. Symmetry factors and signs are not shown for better readability. Not shown are the flows of ghost and gluon propagators (see [66, 67]) as well as those of the effective potential.